#7002016

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

1405 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei,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, 6909212);