#6461322

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1882 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=gh,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6368518);