#7156219

order	512
nilpotency class	3
isoclinism	isoclinic class of #7021375
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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1957 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=f,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,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, 7063415);