#72316772

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1889
commutator	C₂×C₂²⋊C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂⁴
prime radical	#6487648	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

3247 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,11)(4,14)(5,10)(6,8)(7,12)(9,13)(15,16)(17,18)(19,23)(20,22)(21,28)(24,29)(25,31)(26,27)(30,32);(1,3)(2,7)(4,5)(6,12)(8,11)(9,10)(13,15)(14,16)(17,19,21,25)(18,23,24,31)(20,27,28,30)(22,26,29,32);(2,8)(3,12)(4,13)(10,16)(17,20,22,18)(19,23,26,27)(21,28,29,24)(25,31,32,30);(1,4,5,3,6,13,15,12)(2,9,10,7,8,14,16,11)(18,20)(19,25)(23,30)(24,28)(26,32)(27,31)

Polycyclic presentation:

2^10|b^2=e,c^2=j,d^2=f,f^2=i,b^a=bg,c^a=cij,c^b=cij,d^a=dij,d^b=defi,d^c=dij,e^a=eh,f^b=fi,g^b=gh,g^d=ghj,g^e=gj,h^b=hj

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316779, #72316802, #72317002
maximal normal subgroup of	#72316779, #72316802, #72317002

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.6*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);