#6489740

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2751 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)(7,11)(8,12)(9,13)(10,14)(15,18)(16,19)(17,20)(21,22,27,33)(23,30,37,43)(24,38,41,46)(25,31,26,32)(28,35,44,36)(29,40,47,39)(34,48,51,42)(45,49,52,50);(2,4)(5,7,10,17)(6,11,14,20)(8,16,9,15)(12,19,13,18)(21,23)(22,28)(24,34)(25,35)(26,36)(27,37)(29,42)(30,31)(32,43)(33,44)(38,52)(39,49)(40,50)(41,51)(45,46)(47,48);(5,8,10,9)(6,12,14,13)(7,15,17,16)(11,18,20,19)(21,24,27,41)(22,29,33,47)(23,34,37,51)(25,39,26,40)(28,42,44,48)(30,45,43,52)(31,46,32,38)(35,49,36,50);(5,9,10,8)(6,13,14,12)(7,16,17,15)(11,19,20,18)(21,23)(22,30)(24,34)(25,36)(26,35)(27,37)(28,31)(29,45)(32,44)(33,43)(38,48)(39,50)(40,49)(41,51)(42,46)(47,52);(6,14)(11,20)(12,13)(18,19)(21,25,27,26)(22,31,33,32)(23,35,37,36)(24,39,41,40)(28,30,44,43)(29,46,47,38)(34,49,51,50)(42,45,48,52)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,d^2=h,e^2=i,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, 6396936);