#7101209

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1413 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:

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

Polycyclic presentation:

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

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, 7008405);