#7018509

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1298 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=fi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6925705);