#7623827

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623613
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	#61262	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #525 ◁ #2929 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1276 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531023);