#6677531

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

1701 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)(21,22,26,24)(23,31,35,28)(25,37,39,29)(27,30,41,38)(32,36,43,34)(33,42,48,45)(40,50,51,44)(46,49,52,47);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,23,26,35)(22,28,24,31)(25,33,39,48)(27,34,41,36)(29,42,37,45)(30,43,38,32)(40,49,51,47)(44,52,50,46);(2,4)(21,24,26,22)(23,32,35,43)(25,37,39,29)(27,30,41,38)(28,34,31,36)(33,46,48,52)(40,44,51,50)(42,49,45,47);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,25)(22,29)(23,33)(24,37)(26,39)(27,40)(28,42)(30,44)(31,45)(32,46)(34,49)(35,48)(36,47)(38,50)(41,51)(43,52);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(23,34)(25,39)(27,41)(28,43)(29,37)(30,38)(31,32)(33,47)(35,36)(42,46)(45,52)(48,49)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,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, 6584727);