#6557968

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1495 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:

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

Polycyclic presentation:

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

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, 6465164);