#6852485

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1492 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)(3,7)(4,6)(5,8)(17,18,23,19)(20,28,30,32)(21,25,22,27)(24,31,26,29);(1,3)(2,5)(4,8)(6,7)(9,10,13,16)(11,15,12,14)(17,19,23,18)(20,24,30,26)(21,25,22,27)(28,31,32,29);(3,8)(5,7)(9,11,13,12)(10,14,16,15)(17,20,23,30)(18,24,19,26)(21,29,22,31)(25,32,27,28);(2,6)(5,7)(17,21,23,22)(18,25,19,27)(20,29,30,31)(24,32,26,28);(9,12,13,11)(10,15,16,14)(18,19)(24,26)(25,27)(28,32)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=fh,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6759681);