#7073316

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	#45385	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1554 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)(11,13)(12,14)(15,16)(17,18,22,27,21,26,36,24)(19,28,31,40,29,23,37,38)(20,32,35,41,34,25,42,44)(30,39,45,47,33,43,46,48);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,33)(21,29)(22,31)(24,40)(25,43)(26,28)(27,38)(30,34)(32,39)(35,46)(36,37)(41,47)(42,45)(44,48);(9,11,12,16)(10,13,14,15)(18,24)(19,29)(20,34)(22,36)(23,38)(25,44)(26,27)(28,40)(32,41)(39,48)(43,47)(45,46);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20,21,34)(18,25,26,32)(19,30,29,33)(22,35,36,42)(23,39,28,43)(24,41,27,44)(31,45,37,46)(38,47,40,48);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(18,26)(20,34)(23,28)(24,27)(30,33)(35,42)(38,40)(45,46)

Polycyclic presentation:

2^9|a^2=fgi,c^2=hi,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=ehi,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, 6980512);