#7182799

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1691 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

(1,2,3,4)(8,16)(11,17)(15,20)(18,19)(21,22,27,31)(23,33,25,30)(24,29,37,46)(26,41,39,32)(28,40,43,35)(34,47,36,44)(38,51,49,45)(42,50,52,48);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,26,35,27,25,39,40)(22,28,32,33,31,43,41,30)(24,34,38,48,37,36,49,50)(29,42,45,47,46,52,51,44);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24,27,37)(22,29,31,46)(23,34,25,36)(26,38,39,49)(28,42,43,52)(30,44,33,47)(32,45,41,51)(35,48,40,50);(7,14)(10,13)(15,20)(18,19)(21,25,27,23)(22,30,31,33)(24,36,37,34)(26,35,39,40)(28,32,43,41)(29,44,46,47)(38,48,49,50)(42,45,52,51);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(22,31)(24,37)(28,43)(30,33)(32,41)(34,36)(38,49)(48,50)

Polycyclic presentation:

2^9|a^2=gi,b^2=fh,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,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, 7089995);