#6716485

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	#42504	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1418 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18,20,23)(19,22,27,31)(21,24,29,30)(25,28,32,26);(1,3,5,4)(2,6,7,8)(10,13)(11,15)(17,19,20,27)(18,22,23,31)(21,26,29,28)(24,25,30,32);(2,7)(6,8)(9,11,12,15)(10,14,13,16)(17,18)(19,25)(20,23)(21,30)(22,26)(24,29)(27,32)(28,31);(10,13)(14,16)(18,23)(22,31)(24,30)(25,32);(1,4,5,3)(2,8,7,6)(19,26)(21,29)(22,25)(24,30)(27,28)(31,32)

Polycyclic presentation:

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

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, 6623681);