#7623943

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623626
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #533 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1222 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,d^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531139);