#7615555

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1320 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,30)(21,23)(22,26)(25,32)(27,40)(28,34)(29,37)(31,38)(33,35)(36,42)(39,44)(41,46)(43,45)(47,48);(2,6)(7,8)(11,16)(13,15)(18,23)(20,31)(22,33)(24,34)(25,38)(27,41)(29,43)(30,39)(32,44)(36,46)(40,47)(42,48);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,27)(21,28)(22,29)(23,34)(25,36)(26,37)(30,40)(31,41)(32,42)(33,43)(35,45)(38,46)(39,47)(44,48);(10,14)(13,15)(17,20,22,31,21,32,33,44)(18,25,26,39,23,30,35,38)(19,27,29,41,28,42,43,48)(24,36,37,47,34,40,45,46);(1,4,5,3)(2,7,6,8)(18,23)(19,28)(20,32)(26,35)(29,43)(31,44)(36,40)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7522751);