#7148692

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

1423 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)(17,18,19,27,22,28,23,24)(20,31,36,25,33,37,29,40)(21,26,30,41,35,42,34,38)(32,44,45,39,46,47,43,48);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(17,19,22,23)(18,24,28,27)(20,29,33,36)(21,34,35,30)(25,37,40,31)(26,41,42,38)(32,45,46,43)(39,44,48,47);(9,10)(11,14)(12,13)(15,16)(17,20,22,33)(18,25,28,40)(19,29,23,36)(21,32,35,46)(24,37,27,31)(26,39,42,48)(30,43,34,45)(38,47,41,44);(3,8)(5,7)(10,13)(14,16)(17,21)(18,26)(19,30)(20,32)(22,35)(23,34)(24,38)(25,39)(27,41)(28,42)(29,43)(31,44)(33,46)(36,45)(37,47)(40,48);(9,11,12,15)(10,14,13,16)(17,19,22,23)(18,27,28,24)(20,29,33,36)(21,34,35,30)(25,31,40,37)(26,38,42,41)(32,45,46,43)(39,47,48,44)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,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, 7055888);