#6432387

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 72 points:

(1,2)(3,6)(4,5)(7,8)(9,10,14,20,16,22,15,21)(11,17,25,39,26,40,36,48)(12,27,30,44,32,43,31,18)(13,19,34,46,33,45,35,47)(23,41,28,54,50,64,53,37)(24,38,52,63,51,62,60,68)(29,55,58,66,57,67,59,42)(49,65,56,71,69,72,70,61);(2,5)(6,8)(9,11)(10,17)(12,28)(13,24)(14,25)(15,36)(16,26)(18,41)(19,38)(20,39)(21,48)(22,40)(23,31)(27,54)(29,56)(30,50)(32,53)(33,51)(34,52)(35,60)(37,43)(42,65)(44,64)(45,62)(46,63)(47,68)(49,59)(55,71)(57,70)(58,69)(61,67)(66,72);(9,12,16,32)(10,18,22,44)(11,23,26,50)(13,29,33,57)(14,31,15,30)(17,37,40,54)(19,42,45,66)(20,43,21,27)(24,49,51,69)(25,53,36,28)(34,59,35,58)(38,61,62,71)(39,64,48,41)(46,67,47,55)(52,70,60,56)(63,72,68,65);(1,3,4,7)(2,6,5,8)(9,13)(10,19)(11,24)(12,29)(14,34)(15,35)(16,33)(17,38)(18,42)(20,46)(21,47)(22,45)(23,49)(25,52)(26,51)(27,55)(28,56)(30,58)(31,59)(32,57)(36,60)(37,61)(39,63)(40,62)(41,65)(43,67)(44,66)(48,68)(50,69)(53,70)(54,71)(64,72);(13,33)(19,45)(24,51)(29,57)(34,35)(38,62)(42,66)(46,47)(49,69)(52,60)(55,67)(56,70)(58,59)(61,71)(63,68)(65,72)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339583);