#6542936

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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

1582 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6450132);