#7064164

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1427 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)(9,10)(11,13)(12,14)(15,16)(17,18,24,39,23,29,22,28)(19,30,33,43,34,25,35,42)(20,27,36,26,21,38,37,44)(31,46,48,45,32,41,47,40);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,31)(21,32)(22,35)(23,34)(24,33)(26,40)(27,41)(28,43)(29,30)(36,48)(37,47)(38,46)(39,42)(44,45);(9,11,12,16)(10,13,14,15)(17,20,23,21)(18,26,29,44)(19,31,34,32)(22,36,24,37)(25,40,30,45)(27,28,38,39)(33,47,35,48)(41,43,46,42);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,21,23,20)(18,27,29,38)(19,32,34,31)(22,36,24,37)(25,41,30,46)(26,28,44,39)(33,47,35,48)(40,43,45,42);(10,14)(13,15)(17,22,23,24)(18,28,29,39)(19,33,34,35)(20,36,21,37)(25,42,30,43)(26,38,44,27)(31,47,32,48)(40,41,45,46)

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,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, 6971360);