#6811721

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1640 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)(5,6,10,13)(7,14,16,11)(8,17,9,12)(15,19,18,20);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,22,26,30)(23,31,32,45)(24,35,38,50)(25,39,36,29)(27,40,41,34)(28,43,44,37)(33,48,47,42)(46,51,52,49);(2,4)(21,23,25,34,26,32,36,40)(22,27,29,31,30,41,39,45)(24,33,37,49,38,47,43,51)(28,42,35,46,44,48,50,52);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(23,32)(27,41)(31,45)(33,47)(34,40)(42,48)(46,52)(49,51);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,24,36,43,26,38,25,37)(22,28,39,50,30,44,29,35)(23,33,40,51,32,47,34,49)(27,42,45,52,41,48,31,46)

Polycyclic presentation:

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

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, 6718917);