#6481721

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470161
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2865 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)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22)(23,33)(24,38)(25,32)(26,31)(27,30)(28,35)(29,40)(34,46)(36,44)(37,45)(39,43)(41,47)(42,48)(49,50)(51,52);(2,4)(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,23,27,37)(22,28,30,44)(24,39,42,34)(25,36,26,35)(29,46,47,43)(31,45,32,33)(38,49,48,52)(40,50,41,51);(7,13)(10,17)(14,15)(18,19)(21,24)(22,29)(23,34)(25,40)(26,41)(27,42)(28,43)(30,47)(31,38)(32,48)(33,49)(35,50)(36,51)(37,39)(44,46)(45,52);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(22,30)(28,44)(29,47)(31,32)(33,45)(38,48)(43,46)(49,52);(6,12)(10,17)(11,20)(18,19)(21,25,27,26)(22,31,30,32)(23,35,37,36)(24,40,42,41)(28,33,44,45)(29,38,47,48)(34,50,39,51)(43,49,46,52)

Polycyclic presentation:

2^9|b^2=i,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6388917);