#6908043

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

2146 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,25)(19,26,24,21)(20,23,28,30)(27,32,31,29);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,16)(14,15)(20,28)(23,30)(27,31)(29,32);(1,2)(3,5)(4,6)(7,8)(11,16)(13,15)(17,19,21,18,22,24,26,25)(20,27,29,23,28,31,32,30);(2,6)(5,7)(10,14)(13,15)(17,20,22,28)(18,23,25,30)(19,27,24,31)(21,29,26,32);(3,8)(5,7)(20,28)(23,30)(27,31)(29,32)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^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=efh,d^c=dfh,e^d=ei

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, 6815239);