#7181151

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1694 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)(21,22,25,30)(23,32,34,27)(24,29,36,43)(26,28,38,31)(33,45,46,40)(35,41,48,39)(37,42,49,44)(47,51,52,50);(2,3)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,25,34)(22,27,30,32)(24,33,36,46)(26,35,38,48)(28,39,31,41)(29,40,43,45)(37,47,49,52)(42,50,44,51);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(22,28)(26,38)(27,39)(29,42)(30,31)(32,41)(35,48)(37,49)(40,50)(43,44)(45,51)(47,52);(1,3,4,2)(7,14)(10,13)(15,20)(18,19)(21,24)(22,29)(23,33)(25,36)(26,37)(27,40)(28,42)(30,43)(31,44)(32,45)(34,46)(35,47)(38,49)(39,50)(41,51)(48,52);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(22,30)(24,36)(27,32)(28,31)(33,46)(37,49)(39,41)(47,52)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7088347);