#6557824

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535641
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	#25390	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1597 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)(17,18,22,27)(19,28,31,24)(20,23,30,32)(21,25,29,26);(2,6)(7,8)(9,11,13,10)(12,15,16,14)(17,19,29,30,22,31,21,20)(18,23,25,24,27,32,26,28);(1,3,5,4)(2,7,6,8)(10,11)(14,15)(17,20,21,31,22,30,29,19)(18,24,26,23,27,28,25,32);(1,4,5,3)(2,8,6,7)(12,16)(14,15)(17,21,22,29)(18,25,27,26)(19,30,31,20)(23,28,32,24);(9,12,13,16)(10,14,11,15)(17,21,22,29)(18,25,27,26)(19,20,31,30)(23,24,32,28)

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=f,d^2=fgh,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,d^c=di,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, 6465020);