#6498669

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1710 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)(7,10)(8,15)(9,11)(12,16)(13,17)(14,20)(18,19)(21,22,26,30,27,31,41,46)(23,32,35,43,36,45,40,28)(24,37,38,52,25,29,39,44)(33,48,49,42,34,47,50,51);(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23,27,36)(22,28,31,43)(24,33,25,34)(26,40,41,35)(29,42,37,51)(30,45,46,32)(38,50,39,49)(44,48,52,47);(1,2,3,4)(6,11)(7,13)(12,15)(14,19)(21,24,27,25)(22,29,31,37)(23,33,36,34)(26,38,41,39)(28,42,43,51)(30,44,46,52)(32,47,45,48)(35,49,40,50);(7,13)(10,17)(14,19)(18,20)(21,25,27,24)(22,29,31,37)(23,34,36,33)(26,39,41,38)(28,42,43,51)(30,44,46,52)(32,47,45,48)(35,50,40,49);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(24,25)(29,37)(33,34)(38,39)(42,51)(44,52)(47,48)(49,50)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,e^c=ei,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, 6405865);