#417719

order	512
nilpotency class	3
isoclinism	isoclinic class of #417718
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15257
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2299 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)(4,20)(5,25)(6,14)(7,12)(8,13)(9,18)(10,23)(11,28)(16,31)(17,32)(19,41)(21,35)(22,37)(24,36)(26,43)(27,40)(29,39)(30,33)(34,47)(38,48)(42,46)(44,62)(45,50)(49,56)(51,54)(52,58)(53,57)(55,61)(59,64)(60,63);(1,3)(2,9)(4,16)(5,26)(6,30)(7,18)(8,19)(10,34)(11,38)(12,41)(13,33)(14,15)(17,27)(20,50)(21,52)(22,46)(23,45)(24,47)(25,56)(28,49)(29,48)(31,37)(32,39)(35,60)(36,42)(40,43)(44,53)(51,59)(54,63)(55,64)(57,58)(61,62);(1,4)(2,10)(3,16)(5,21)(6,22)(7,23)(8,24)(9,31)(11,35)(12,36)(13,37)(14,20)(15,42)(17,44)(18,45)(19,47)(25,57)(26,52)(27,53)(28,54)(29,55)(30,46)(32,58)(33,34)(38,62)(39,51)(40,61)(41,50)(43,60)(48,64)(49,63)(56,59);(1,5,8,29)(2,11,13,40)(3,17,19,49)(4,21,24,55)(6,28,7,27)(9,32,33,56)(10,35,37,61)(12,25,14,39)(15,43,41,38)(16,44,47,63)(18,48,30,26)(20,51,36,57)(22,54,23,53)(31,58,34,59)(42,60,50,62)(45,64,46,52)

Polycyclic presentation:

2^9|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,f^c=fi,e^d=ei,h^d=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286994, #72287261, #72287292
maximal normal subgroup of	#72286994, #72287261, #72287292

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324915);