#6622303

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1450 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)(9,10,13,14)(11,16,12,15)(18,22)(19,25)(21,27)(23,30)(26,31)(28,32);(2,5)(6,8)(10,14)(11,12)(17,18)(19,26)(20,22)(21,24)(23,28)(25,30)(27,29)(31,32);(9,11,13,12)(10,15,14,16)(17,19,27,32,20,28,21,25)(18,23,29,30,22,26,24,31);(1,3,4,7)(2,6,5,8)(11,12)(15,16)(18,22)(19,28)(24,29)(25,32);(9,12,13,11)(10,15,14,16)

Polycyclic presentation:

2^9|a^2=fi,c^2=gi,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^d=ehi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6529499);