#7617660

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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1345 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)(3,6)(4,5)(7,8)(10,14)(11,12)(17,18)(19,24)(20,33)(21,27)(22,28)(23,29)(25,32)(26,37)(30,42)(31,41)(34,47)(35,40)(36,46)(38,45)(39,43)(44,48);(2,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19,22,32)(18,24,28,25)(20,34,37,48)(21,31,23,39)(26,44,33,47)(27,41,29,43)(30,36,35,38)(40,45,42,46);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,19)(18,25)(20,35)(21,39)(22,32)(23,31)(24,28)(26,40)(27,41)(29,43)(30,37)(33,42)(34,36)(38,48)(44,46)(45,47);(9,11,13,12)(10,15,14,16)(17,20,21,36,22,37,23,38)(18,26,27,45,28,33,29,46)(19,30,39,48,32,35,31,34)(24,40,43,47,25,42,41,44);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,27,28,29)(19,31,32,39)(20,36,37,38)(24,41,25,43)(26,45,33,46)(30,34,35,48)(40,44,42,47)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=hi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,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, 7524856);