#6537043

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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 ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1627 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)(21,22,27,31)(23,32,34,46)(24,29,37,43)(25,30,26,40)(28,35,42,36)(33,45,49,52)(38,44,39,50)(41,47,51,48);(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23,25,35,27,34,26,36)(22,28,40,46,31,42,30,32)(24,33,38,47,37,49,39,48)(29,41,50,52,43,51,44,45);(2,4)(6,10)(11,17)(12,18)(19,20)(21,24)(22,29)(23,33)(25,38)(26,39)(27,37)(28,41)(30,44)(31,43)(32,45)(34,49)(35,47)(36,48)(40,50)(42,51)(46,52);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(23,34)(24,37)(28,42)(29,43)(32,46)(35,36)(38,39)(44,50);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(21,25,27,26)(22,30,31,40)(23,35,34,36)(24,38,37,39)(28,32,42,46)(29,44,43,50)(33,47,49,48)(41,45,51,52)

Polycyclic presentation:

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

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, 6444239);