#6434096

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432833
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1973 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,31)(21,25)(22,26)(23,30)(24,29)(27,32);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,20)(18,23,26,24)(21,29,32,30)(25,28,27,31);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,20,22,19)(18,24,26,23)(21,30,32,29)(25,31,27,28);(10,14)(13,16)(17,21,22,32)(18,25,26,27)(19,29,20,30)(23,28,24,31);(18,26)(23,24)(25,27)(28,31)

Polycyclic presentation:

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

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, 6341292);