#7018256

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1300 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)(17,18,23,37,22,28,21,27)(19,24,31,40,33,42,32,41)(20,26,34,25,36,43,35,44)(29,45,47,38,30,39,46,48);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,33)(18,24,28,42)(20,29,36,30)(21,32,23,31)(25,38,44,48)(26,45,43,39)(27,41,37,40)(34,47,35,46);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,29)(21,34)(22,36)(23,35)(24,38)(26,37)(27,43)(28,44)(30,33)(31,46)(32,47)(39,41)(40,45)(42,48);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20)(18,26)(19,30)(21,35)(22,36)(23,34)(24,39)(25,37)(27,44)(28,43)(29,33)(31,46)(32,47)(38,41)(40,48)(42,45);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,21,22,23)(18,27,28,37)(19,31,33,32)(20,34,36,35)(24,40,42,41)(25,43,44,26)(29,46,30,47)(38,45,48,39)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=f,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,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, 6925452);