#6436563

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432843
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3455	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2946 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)(8,12)(9,11)(13,16)(14,17)(15,18)(19,20)(21,22,27,32)(23,33,36,48)(24,29,39,46)(25,31,26,30)(28,41,43,35)(34,49,37,51)(38,47,40,45)(42,52,44,50);(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20)(21,23)(22,28)(24,37)(25,35)(26,41)(27,36)(29,44)(30,33)(31,48)(32,43)(34,39)(38,52)(40,50)(42,46)(45,51)(47,49);(2,4)(7,13)(10,16)(14,19)(17,20)(21,24)(22,29)(23,34)(25,38)(26,40)(27,39)(28,42)(30,45)(31,47)(32,46)(33,49)(35,50)(36,37)(41,52)(43,44)(48,51);(6,11)(10,16)(12,18)(17,20)(21,25,27,26)(22,30,32,31)(23,35,36,41)(24,38,39,40)(28,33,43,48)(29,45,46,47)(34,50,37,52)(42,49,44,51);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,17,16,20)(24,39)(29,46)(34,37)(38,40)(42,44)(45,47)(49,51)(50,52)

Polycyclic presentation:

2^9|a^2=i,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,f^b=fi,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, 6343759);