#6808637

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1682 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:

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

Polycyclic presentation:

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

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, 6715833);