#72233045

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	C₂×#62817	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

25883 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^10|d^2=i,e^2=j,f^2=ij,d^a=dj,d^b=dg,d^c=dh,e^d=ef,f^b=fi,f^c=fj,f^d=fij,f^e=fij,g^e=gi,h^e=hj

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234695, #72284244, #72284573, …
maximal normal subgroup of	#72234695, #72284244, #72284573, …

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, 4, F.9);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.6);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);