#6992857

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
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	#39130	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1543 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fhi,d^2=fhi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6900053);