#6564610

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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

1825 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)(21,22,27,31)(23,32,36,43)(24,29,40,48)(25,30,26,41)(28,35,44,34)(33,49,37,52)(38,46,39,47)(42,50,45,51);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,26,34,27,36,25,35)(22,28,30,32,31,44,41,43)(24,37,38,50,40,33,39,51)(29,45,47,52,48,42,46,49);(2,4)(7,13)(10,16)(14,19)(17,20)(21,24)(22,29)(23,33)(25,39)(26,38)(27,40)(28,42)(30,47)(31,48)(32,49)(34,51)(35,50)(36,37)(41,46)(43,52)(44,45);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(21,25,27,26)(22,30,31,41)(23,34,36,35)(24,38,40,39)(28,43,44,32)(29,46,48,47)(33,50,37,51)(42,49,45,52);(8,15)(11,18)(14,19)(17,20)(21,25,27,26)(22,30,31,41)(23,35,36,34)(24,39,40,38)(28,32,44,43)(29,47,48,46)(33,50,37,51)(42,49,45,52)

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=i,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=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6471806);