#6477266

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
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	#2615	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1612 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6384462);