#6684474

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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

1736 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)(8,12)(9,18)(10,13)(11,17)(15,20)(16,19)(21,22,27,31)(23,28,26,41)(24,34,37,50)(25,30,39,46)(29,32,44,36)(33,42,40,52)(35,49,38,51)(43,47,45,48);(1,2)(3,4)(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(21,23,27,26)(22,28,31,41)(24,32,37,36)(25,33,39,40)(29,34,44,50)(30,42,46,52)(35,47,38,48)(43,49,45,51);(2,4)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(21,24,26,36,27,37,23,32)(22,29,28,34,31,44,41,50)(25,38,40,47,39,35,33,48)(30,45,42,51,46,43,52,49);(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(21,25)(22,30)(23,33)(24,35)(26,40)(27,39)(28,42)(29,43)(31,46)(32,47)(34,49)(36,48)(37,38)(41,52)(44,45)(50,51);(2,4)(7,17)(11,14)(15,16)(19,20)(25,39)(30,46)(33,40)(35,38)(42,52)(43,45)(47,48)(49,51)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,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, 6591670);