#7623778

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	#61084	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1530 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:

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

Polycyclic presentation:

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

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, 7530974);