#7065261

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

1781 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)(11,21,23,35)(12,26,30,39)(13,31)(14,20)(15,19)(16,24,36,25)(17,29,38,28)(18,33)(22,37)(27,34)(32,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,27)(13,28)(14,25)(15,23)(17,37)(18,39)(19,36)(20,35)(21,40)(22,30)(24,32)(26,31)(29,33)(34,38);(1,3,4,8)(2,5,6,7)(9,12)(10,17)(11,22)(13,24)(14,28)(15,30)(16,34)(18,21)(19,38)(20,39)(23,27)(25,33)(26,40)(29,32)(31,35)(36,37);(3,8)(5,7)(9,13,32,22,15,33,14,27)(10,18,40,34,19,31,20,37)(11,12,25,29,23,30,24,28)(16,17,35,26,36,38,21,39);(2,6)(5,7)(10,19)(11,23)(13,33)(17,38)(20,40)(22,27)(24,25)(26,39)

Polycyclic presentation:

2^9|a^2=fgi,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, 6972457);