#6481729

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2918 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6388925);