#7594412

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1419 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)(3,5)(4,6)(7,8)(17,18,23,27)(19,28,32,45)(20,25,35,43)(21,26,22,37)(24,30,39,31)(29,44,34,48)(33,40,36,42)(38,47,41,46);(1,3)(2,5)(4,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,30,23,32,22,31)(18,24,37,45,27,39,26,28)(20,29,36,47,35,34,33,46)(25,38,40,48,43,41,42,44);(2,6)(3,7)(9,11,12,15)(10,13,14,16)(17,20)(18,25)(19,29)(21,36)(22,33)(23,35)(24,38)(26,42)(27,43)(28,44)(30,47)(31,46)(32,34)(37,40)(39,41)(45,48);(2,6)(5,8)(10,14)(13,16)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,33,35,36)(24,28,39,45)(25,40,43,42)(29,46,34,47)(38,48,41,44);(17,19,21,30,23,32,22,31)(18,24,37,45,27,39,26,28)(20,34,36,46,35,29,33,47)(25,41,40,44,43,38,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=h,d^2=i,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=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, 7501608);