#6682947

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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	#35737	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1994 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)(15,19,24,34)(16,27,29,20)(17,21,31,39)(23,33,26,36)(25,42,41,35)(28,44,43,37)(30,40,32,38);(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,28)(17,23)(18,24)(20,37)(21,33)(22,34)(25,30)(26,31)(27,44)(29,43)(32,41)(35,38)(36,39)(40,42);(1,2,3,4)(15,23)(17,31)(19,33)(21,39)(24,26)(25,28)(30,32)(34,36)(35,37)(38,40)(41,43)(42,44);(6,10)(9,11)(15,24)(16,29)(19,34)(20,27)(23,26)(30,32)(33,36)(38,40);(7,12)(9,11)(13,16,18,29)(14,20,22,27)(15,25,24,41)(17,32,31,30)(19,35,34,42)(21,40,39,38)(23,28,26,43)(33,37,36,44)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6590143);