#6677320

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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

1714 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)(23,28)(24,29)(25,30)(26,32)(27,31)(33,44)(34,36)(35,48)(37,45)(38,46)(39,43)(40,42)(41,47)(49,52)(50,51);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,23)(22,28)(24,38)(25,34)(26,35)(27,37)(29,46)(30,36)(31,45)(32,48)(33,40)(39,52)(41,51)(42,44)(43,49)(47,50);(2,4)(21,24,27,41)(22,29,31,47)(23,33,37,49)(25,42,32,39)(26,43,30,40)(28,44,45,52)(34,50,48,46)(35,38,36,51);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,25,27,32)(22,30,31,26)(23,34,37,48)(24,39,41,42)(28,36,45,35)(29,43,47,40)(33,46,49,50)(38,52,51,44);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(22,31)(23,35)(26,30)(28,34)(29,47)(33,38)(36,37)(40,43)(44,50)(45,48)(46,52)(49,51)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6584516);