#7083106

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

1372 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=gh,d^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,f^d=fi

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, 6990302);