#7594538

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	#24778	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1397 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=hi,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, 7501734);