#6832532

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1395 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=fhi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6739728);