#7143400

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

1528 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)(19,24)(20,30)(21,26)(22,35)(23,27)(25,28)(29,38)(31,44)(32,37)(33,41)(34,42)(36,40)(39,43)(45,48)(46,47);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(17,19,23,22)(18,24,27,35)(20,28,32,36)(21,29,34,33)(25,37,40,30)(26,38,42,41)(31,43,46,45)(39,47,48,44);(9,10)(11,14)(12,13)(15,16)(17,20,23,32)(18,25,27,40)(19,28,22,36)(21,31,34,46)(24,37,35,30)(26,39,42,48)(29,43,33,45)(38,47,41,44);(3,8)(5,7)(10,13)(14,16)(17,21)(18,26)(19,29)(20,31)(22,33)(23,34)(24,38)(25,39)(27,42)(28,43)(30,44)(32,46)(35,41)(36,45)(37,47)(40,48);(9,11,12,15)(10,14,13,16)(17,19,23,22)(18,24,27,35)(20,28,32,36)(21,33,34,29)(25,37,40,30)(26,41,42,38)(31,45,46,43)(39,44,48,47)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,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, 7050596);