#7065253

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

1726 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=hi,d^2=fi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 6972449);