#7181129

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1678 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=i,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,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, 7088325);