#6569666

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3347	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2040 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:

(2,4)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22,27,31)(23,32,34,47)(24,29,37,43)(25,30,26,40)(28,35,42,36)(33,46,49,51)(38,44,39,45)(41,50,52,48);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,26,36,27,34,25,35)(22,28,30,32,31,42,40,47)(24,33,38,48,37,49,39,50)(29,41,45,46,43,52,44,51);(1,2,3,4)(7,13)(10,16)(14,19)(17,20)(21,24,27,37)(22,29,31,43)(23,33,34,49)(25,39,26,38)(28,41,42,52)(30,45,40,44)(32,46,47,51)(35,50,36,48);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(23,34)(24,37)(28,42)(29,43)(32,47)(35,36)(38,39)(44,45);(8,15)(11,18)(14,19)(17,20)(21,25,27,26)(22,30,31,40)(23,35,34,36)(24,38,37,39)(28,32,42,47)(29,44,43,45)(33,48,49,50)(41,51,52,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=gi,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^c=ei,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, 6476862);