#6832579

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806903
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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1412 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fhi,d^2=i,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,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, 6739775);