#6816198

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1768 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=ghi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6723394);