#6565897

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1695 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)(5,6,10,9)(7,11,13,15)(8,12,16,17)(14,18,19,20)(21,22)(23,32)(24,29)(25,30)(26,40)(27,31)(28,34)(33,46)(35,42)(36,44)(37,45)(38,50)(39,43)(41,48)(47,52)(49,51);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,19)(17,20)(21,23,26,34,27,35,25,39)(22,28,30,32,31,43,40,42)(24,33,38,48,36,47,37,49)(29,41,45,46,44,51,50,52);(2,4)(7,13)(11,15)(14,19)(18,20)(21,24)(22,29)(23,33)(25,37)(26,38)(27,36)(28,41)(30,45)(31,44)(32,46)(34,48)(35,47)(39,49)(40,50)(42,52)(43,51);(5,8,10,16)(6,12,9,17)(7,14,13,19)(11,18,15,20)(24,36)(29,44)(33,47)(37,38)(41,51)(45,50)(46,52)(48,49);(5,9,10,6)(7,15,13,11)(8,12,16,17)(14,18,19,20)(21,25,27,26)(22,30,31,40)(23,34,35,39)(24,37,36,38)(28,42,43,32)(29,45,44,50)(33,48,47,49)(41,52,51,46)

Polycyclic presentation:

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

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, 6473093);