#6536576

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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	#23390	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1843 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)(4,5)(7,8)(17,18,24,21)(19,27,30,31)(20,22,26,23)(25,28,32,29);(9,10)(11,14)(12,13)(15,16)(17,19,23,29,24,30,22,28)(18,25,26,27,21,32,20,31);(2,5)(6,8)(10,13)(14,16)(17,20)(18,23)(19,27)(21,22)(24,26)(25,28)(29,32)(30,31);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(17,21,24,18)(19,25,30,32)(20,22,26,23)(27,28,31,29);(11,15)(14,16)(17,22,24,23)(18,26,21,20)(19,28,30,29)(25,27,32,31)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6443772);