#72334063

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁶
commutator	C₂³	C₂⁵×C₄
prime radical	C₂×#35346	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

29129 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^10|b^2=h,d^2=hj,b^a=bj,c^b=cg,d^b=dj,d^c=di,e^b=ej,e^d=ej,f^a=fi,f^b=fj,f^c=fi,f^d=fgi,f^e=fgij

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334064
maximal normal subgroup of	#72334064

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