#6810558

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#29630	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2052 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(5,6)(7,11)(8,10)(9,12)(13,14,17,22)(15,24,26,19)(16,28,29,21)(18,20,31,23)(25,33,38,37)(27,34,40,32)(30,36,41,35)(39,42,44,43);(2,3)(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,25)(17,26)(18,27)(20,32)(21,33)(22,24)(23,34)(28,37)(29,38)(30,39)(31,40)(35,42)(36,43)(41,44);(1,2,4,3)(7,12)(9,11)(14,20)(18,31)(19,32)(21,35)(22,23)(24,34)(27,40)(28,36)(30,41)(33,42)(37,43)(39,44);(6,10)(9,11)(13,16,17,29)(14,21,22,28)(15,25,26,38)(18,30,31,41)(19,33,24,37)(20,35,23,36)(27,39,40,44)(32,42,34,43);(1,3,4,2)(15,26)(16,29)(19,24)(21,28)(27,40)(30,41)(32,34)(35,36)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fhi,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6717754);