#6436689

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432843
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×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1793 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:

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

Polycyclic presentation:

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

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, 6343885);