#6436596

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432861
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2868 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^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,d^c=di,e^d=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, 6343792);