#7620482

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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1274 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fi,d^2=fghi,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei

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, 7527678);