#7014357

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=fi,d^2=i,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=eh,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, 6921553);