#6808628

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1668 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)(21,22,26,30)(23,31,33,47)(24,36,39,51)(25,40,32,29)(27,41,42,35)(28,44,46,38)(34,45,37,43)(48,52,50,49);(2,4)(21,23,32,41,26,33,25,35)(22,27,40,47,30,42,29,31)(24,37,44,49,39,34,38,52)(28,45,51,48,46,43,36,50);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(23,33)(27,42)(31,47)(34,37)(35,41)(43,45)(48,50)(49,52);(7,15)(10,13)(14,20)(18,19)(21,24,32,44,26,39,25,38)(22,28,40,51,30,46,29,36)(23,34,41,52,33,37,35,49)(27,43,47,50,42,45,31,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=f,e^2=hi,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, 6715824);