#7010963

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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 ◁ #2395 ◁ 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)(17,18,21,28,22,27,37,26)(19,29,31,41,32,23,38,42)(20,25,36,44,35,46,34,24)(30,45,48,39,33,40,47,43);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,32)(18,23,27,29)(20,33,35,30)(21,31,37,38)(24,43,44,39)(25,45,46,40)(26,41,28,42)(34,48,36,47);(9,11)(10,13)(12,16)(14,15)(17,20,22,35)(18,24,27,44)(19,30,32,33)(21,34,37,36)(23,39,29,43)(25,28,46,26)(31,47,38,48)(40,42,45,41);(3,8)(6,7)(11,16)(13,15)(17,20)(18,25)(19,30)(21,36)(22,35)(23,40)(24,26)(27,46)(28,44)(29,45)(31,48)(32,33)(34,37)(38,47)(39,41)(42,43);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,21,22,37)(18,26,27,28)(19,31,32,38)(20,34,35,36)(23,41,29,42)(24,25,44,46)(30,47,33,48)(39,40,43,45)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6918159);