#6557961

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	#25392	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1584 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)(17,18)(19,29)(20,31)(21,26)(22,27)(23,28)(24,30)(25,32);(2,6)(7,8)(9,11,13,10)(12,15,16,14)(17,19,22,30,23,31,21,25)(18,24,26,29,28,32,27,20);(1,3,5,4)(2,7,6,8)(10,11)(14,15)(17,20,22,24,23,29,21,32)(18,25,26,19,28,30,27,31);(1,4,5,3)(2,8,6,7)(12,16)(14,15)(17,21,23,22)(18,26,28,27)(19,25,31,30)(20,24,29,32);(9,12,13,16)(10,14,11,15)(17,21,23,22)(18,26,28,27)(19,30,31,25)(20,32,29,24)

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=f,d^2=fgh,e^2=h,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, 6465157);