#421065

order	512
nilpotency class	3
isoclinism	isoclinic class of #421064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15283
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60368	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1811 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=ef,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286957, #72287261, #72287292
maximal normal subgroup of	#72286957, #72287261, #72287292

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 328261);