#7014283

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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1297 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)(17,18,21,26,22,27,38,45)(19,28,33,41,32,23,31,42)(20,34,37,43,36,25,35,24)(29,44,47,39,30,40,48,46);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,32)(18,23,27,28)(20,29,36,30)(21,33,38,31)(24,39,43,46)(25,44,34,40)(26,42,45,41)(35,48,37,47);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,20)(18,24)(19,29)(21,35)(22,36)(23,39)(25,26)(27,43)(28,46)(30,32)(31,47)(33,48)(34,45)(37,38)(40,41)(42,44);(3,8)(6,7)(11,16)(13,15)(17,20)(18,25)(19,30)(21,37)(22,36)(23,40)(24,26)(27,34)(28,44)(29,32)(31,47)(33,48)(35,38)(39,41)(42,46)(43,45);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,21,22,38)(18,26,27,45)(19,31,32,33)(20,35,36,37)(23,41,28,42)(24,25,43,34)(29,47,30,48)(39,40,46,44)

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^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=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, 6921479);