#7039297

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

1329 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)(11,15)(12,16)(13,14)(17,18,22,28)(19,30,33,24)(20,26,36,45)(21,29,23,27)(25,38,43,35)(31,44,34,40)(32,41,37,46)(39,48,42,47);(1,3)(2,6)(4,7)(5,8)(10,14)(11,12)(17,19,22,33)(18,24,28,30)(20,34,36,31)(21,37,23,32)(25,42,43,39)(26,44,45,40)(27,46,29,41)(35,47,38,48);(9,10)(11,16)(12,15)(13,14)(17,20,22,36)(18,25,28,43)(19,31,33,34)(21,35,23,38)(24,39,30,42)(26,29,45,27)(32,47,37,48)(40,41,44,46);(1,2,5,4)(3,7,8,6)(9,11,13,12)(10,15,14,16)(17,20)(18,26)(19,31)(21,38)(22,36)(23,35)(24,40)(25,27)(28,45)(29,43)(30,44)(32,48)(33,34)(37,47)(39,46)(41,42);(1,4,5,2)(3,7,8,6)(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,27,28,29)(19,32,33,37)(20,35,36,38)(24,41,30,46)(25,26,43,45)(31,47,34,48)(39,44,42,40)

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=i,d^2=fi,e^2=f,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, 6946493);