#6458926

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
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

1828 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)(4,7)(5,8)(9,10,12,14)(11,13,15,16)(17,18)(19,27)(20,24)(21,23)(22,26)(25,30)(28,32)(29,31);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(18,23)(21,30)(24,31)(25,26)(27,32)(28,29);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,19,22,20)(18,24,26,27)(21,28,30,29)(23,31,25,32);(1,4,5,3)(2,7,8,6)(11,15)(13,16)(17,20,22,19)(18,24,26,27)(21,29,30,28)(23,31,25,32);(19,20)(24,27)(28,29)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366122);