#6457461

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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 ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1672 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,18,22,24)(19,21,26,23)(20,25,29,31)(27,30,32,28);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(18,24)(19,26)(25,31)(27,32);(11,15)(13,16)(17,19,23,18,22,26,21,24)(20,27,28,25,29,32,30,31);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,20,22,29)(18,25,24,31)(19,27,26,32)(21,30,23,28);(17,21,22,23)(18,26,24,19)(20,28,29,30)(25,27,31,32)

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei

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, 6364657);