#7182804

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1671 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)(21,22,27,32)(23,33,34,49)(24,29,37,50)(25,30,36,51)(26,41,43,31)(28,44,47,35)(38,52,39,46)(40,45,42,48);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,26,35,27,34,43,44)(22,28,31,33,32,47,41,49)(24,36,38,42,37,25,39,40)(29,48,46,51,50,45,52,30);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24)(22,29)(23,25)(26,38)(27,37)(28,45)(30,33)(31,46)(32,50)(34,36)(35,40)(39,43)(41,52)(42,44)(47,48)(49,51);(7,14)(10,13)(15,20)(18,19)(21,25,27,36)(22,30,32,51)(23,24,34,37)(26,42,43,40)(28,46,47,52)(29,49,50,33)(31,45,41,48)(35,39,44,38);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(23,34)(24,37)(28,47)(29,50)(33,49)(35,44)(38,39)(46,52)

Polycyclic presentation:

2^9|a^2=gi,b^2=fhi,d^2=i,e^2=hi,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, 7090000);