#6564600

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1811 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)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22)(23,32)(24,29)(25,30)(26,40)(27,31)(28,36)(33,45)(34,46)(35,42)(37,43)(38,44)(39,50)(41,48)(47,51)(49,52);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,26,36,27,34,25,35)(22,28,30,32,31,42,40,46)(24,33,39,48,37,49,38,47)(29,41,44,45,43,51,50,52);(2,4)(7,13)(10,16)(14,19)(17,20)(21,24)(22,29)(23,33)(25,38)(26,39)(27,37)(28,41)(30,44)(31,43)(32,45)(34,49)(35,47)(36,48)(40,50)(42,51)(46,52);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(23,34)(24,37)(28,42)(29,43)(32,46)(35,36)(38,39)(44,50);(8,15)(11,18)(14,19)(17,20)(21,25,27,26)(22,30,31,40)(23,35,34,36)(24,38,37,39)(28,32,42,46)(29,44,43,50)(33,47,49,48)(41,45,51,52)

Polycyclic presentation:

2^9|a^2=h,b^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,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, 6471796);