#7369017

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309060
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1402 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)(3,6)(4,5)(7,8)(17,18,23,36,22,28,21,27)(19,24,30,39,32,41,31,40)(20,26,33,25,35,42,34,43)(29,38,46,37,45,48,44,47);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,32)(18,24,28,41)(20,29,35,45)(21,31,23,30)(25,37,43,47)(26,38,42,48)(27,40,36,39)(33,46,34,44);(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,29)(21,33)(22,35)(23,34)(24,37)(26,36)(27,42)(28,43)(30,44)(31,46)(32,45)(38,39)(40,48)(41,47);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20)(18,26)(19,29)(21,34)(22,35)(23,33)(24,38)(25,36)(27,43)(28,42)(30,46)(31,44)(32,45)(37,39)(40,47)(41,48);(2,5)(6,8)(10,14)(13,15)(17,21,22,23)(18,27,28,36)(19,30,32,31)(20,33,35,34)(24,39,41,40)(25,42,43,26)(29,44,45,46)(37,38,47,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,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, 7276213);