#6912694

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907909
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#29607	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1663 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,6)(3,7,8,4)(17,18,21,25)(19,27,29,23)(20,31,33,47)(22,35,32,26)(24,38,39,34)(28,42,44,46)(30,37,43,41)(36,45,40,48);(2,6)(4,7)(9,10)(11,15)(12,14)(13,16)(17,19,21,29)(18,23,25,27)(20,28,33,44)(22,30,32,43)(24,36,39,40)(26,37,35,41)(31,46,47,42)(34,45,38,48);(1,3)(2,7)(4,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,20,32,38,21,33,22,34)(18,24,35,47,25,39,26,31)(19,28,43,48,29,44,30,45)(23,36,41,42,27,40,37,46);(2,6)(3,8)(11,16)(13,15)(18,25)(19,29)(20,33)(26,35)(30,43)(34,38)(36,40)(42,46);(1,4,5,7)(2,3,6,8)(10,14)(13,15)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6819890);