#7007217

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996799
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(9,10)(11,20,24,30)(12,17)(13,18)(14,19)(15,22,16,23)(21,32,36,39)(25,31,37,29)(26,33)(27,34)(28,35)(38,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,25)(13,23)(14,24)(16,19)(17,32)(18,30)(20,33)(21,28)(22,26)(27,37)(29,35)(31,40)(34,39)(36,38);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,21)(13,22)(14,24)(15,19)(17,31)(18,30)(20,33)(23,26)(25,28)(27,36)(29,34)(32,40)(35,39)(37,38);(3,8)(5,7)(9,12,26,38,14,27,13,28)(10,17,33,40,19,34,18,35)(11,21,23,37,24,36,22,25)(15,29,30,39,16,31,20,32);(2,6)(5,7)(12,27)(17,34)(21,36)(25,37)(28,38)(29,31)(32,39)(35,40)

Polycyclic presentation:

2^9|a^2=fgi,d^2=fi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,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, 6914413);