#6477729

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470374
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1843 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

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

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, 6384925);