#384299

order	512
nilpotency class	3
isoclinism	isoclinic class of #384297
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14376
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

935 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284899, #72286090, #72286134
maximal normal subgroup of	#72284899, #72286090, #72286134

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 291495);