#6808795

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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

1662 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)(23,31,35,46)(24,36,37,50)(25,39,32,29)(27,40,42,34)(28,43,44,38)(33,47,48,41)(45,51,52,49);(2,4)(21,23,32,40,26,35,25,34)(22,27,39,46,30,42,29,31)(24,33,43,51,37,48,38,49)(28,41,50,52,44,47,36,45);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(24,37)(28,44)(33,48)(36,50)(38,43)(41,47)(45,52)(49,51);(7,15)(10,13)(14,20)(18,19)(21,24,25,38,26,37,32,43)(22,28,29,36,30,44,39,50)(23,33,34,49,35,48,40,51)(27,41,31,45,42,47,46,52)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=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, 6715991);