#7148772

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

1428 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,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, 7055968);