#7594369

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1428 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)(19,28,32,43)(20,25,34,41)(21,26,22,36)(24,30,38,31)(29,42,45,47)(33,39,35,40)(37,46,48,44);(1,3)(2,5)(4,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,30,23,32,22,31)(18,24,36,43,27,38,26,28)(20,29,35,46,34,45,33,44)(25,37,39,47,41,48,40,42);(2,6)(3,7)(9,11,12,15)(10,13,14,16)(17,20)(18,25)(19,29)(21,35)(22,33)(23,34)(24,37)(26,40)(27,41)(28,42)(30,46)(31,44)(32,45)(36,39)(38,48)(43,47);(2,6)(5,8)(10,14)(13,16)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,33,34,35)(24,28,38,43)(25,39,41,40)(29,44,45,46)(37,47,48,42);(17,19,21,30,23,32,22,31)(18,24,36,43,27,38,26,28)(20,29,35,46,34,45,33,44)(25,37,39,47,41,48,40,42)

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

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, 7501565);