#7608887

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1418 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,24,28,33)(20,25,32,34)(22,26,31,35)(27,36,42,44)(29,37,41,45)(30,39,43,38)(40,47,48,46);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(18,23)(20,30)(22,31)(24,33)(25,38)(27,40)(29,41)(32,43)(34,39)(36,46)(42,48)(44,47);(1,4,5,3)(2,6,7,8)(10,14)(11,15)(17,19,21,28)(18,24,23,33)(20,27,32,42)(22,29,31,41)(25,36,34,44)(26,37,35,45)(30,40,43,48)(38,46,39,47);(10,14)(13,16)(17,20,31,43,21,32,22,30)(18,25,35,38,23,34,26,39)(19,27,41,48,28,42,29,40)(24,36,45,46,33,44,37,47);(2,7)(6,8)(18,23)(19,28)(20,32)(26,35)(29,41)(30,43)(36,44)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,d^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei

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, 7516083);