#6548263

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	#38129	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1601 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,g^c=gi,e^d=eh

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, 6455459);