#6675736

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1750 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:

(5,6)(7,14)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20)(21,22,27,33)(23,34,38,29)(24,31,25,41)(26,32,28,43)(30,42,47,40)(35,49,36,45)(37,50,39,46)(44,51,48,52);(1,2)(3,4)(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(21,23,27,38)(22,29,33,34)(24,35,25,36)(26,37,28,39)(30,44,47,48)(31,45,41,49)(32,46,43,50)(40,51,42,52);(2,4)(21,24,27,25)(22,30,33,47)(23,35,38,36)(26,40,28,42)(29,44,34,48)(31,32,41,43)(37,51,39,52)(45,46,49,50);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,25,27,24)(22,31,33,41)(23,36,38,35)(26,40,28,42)(29,45,34,49)(30,32,47,43)(37,51,39,52)(44,46,48,50);(2,4)(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,26,27,28)(22,32,33,43)(23,37,38,39)(24,40,25,42)(29,46,34,50)(30,41,47,31)(35,51,36,52)(44,49,48,45)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6582932);