#6684589

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1712 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)(7,14)(8,12)(9,18)(10,13)(11,17)(15,20)(16,19)(21,22)(23,31)(24,29)(25,28)(26,30)(27,34)(32,43)(33,49)(35,41)(36,52)(37,44)(38,47)(39,45)(40,48)(42,51)(46,50);(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(21,23,25,34)(22,27,28,31)(24,36,37,50)(26,32,39,35)(29,46,44,52)(30,41,45,43)(33,38,51,40)(42,47,49,48);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(23,32)(26,39)(27,41)(30,45)(31,43)(33,50)(34,35)(36,51)(38,40)(42,52)(46,49)(47,48);(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(22,28)(24,37)(27,31)(30,45)(33,51)(36,50)(38,40)(41,43);(7,17)(11,14)(15,16)(19,20)(21,24,25,37)(22,29,28,44)(23,33,34,51)(26,40,39,38)(27,42,31,49)(30,48,45,47)(32,50,35,36)(41,52,43,46)

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6591785);