#7156228

order	512
nilpotency class	3
isoclinism	isoclinic class of #7021363
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

1935 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,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^c=ei,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, 7063424);