#6908564

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	#39342	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1903 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308943, #72308951
maximal normal subgroup of	#72308943, #72308951

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6815760);