#7015107

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

1319 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)(17,18,22,28,23,29,24,30)(19,31,34,43,35,25,36,42)(20,37,39,26,21,27,38,44)(32,41,48,45,33,46,47,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,35)(18,25,29,31)(20,32,21,33)(22,34,24,36)(26,40,44,45)(27,41,37,46)(28,42,30,43)(38,47,39,48);(9,11)(10,13)(12,16)(14,15)(17,20,23,21)(18,26,29,44)(19,32,35,33)(22,38,24,39)(25,40,31,45)(27,28,37,30)(34,47,36,48)(41,42,46,43);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,21,23,20)(18,27,29,37)(19,33,35,32)(22,38,24,39)(25,41,31,46)(26,28,44,30)(34,47,36,48)(40,42,45,43);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,22,23,24)(18,28,29,30)(19,34,35,36)(20,38,21,39)(25,42,31,43)(26,37,44,27)(32,47,33,48)(40,46,45,41)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6922303);