#499310

order	512
nilpotency class	4
isoclinism	isoclinic class of #13469
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2065
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2065
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

749 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,d^2=i,e^2=h,f^2=h,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gh,h^a=hi,c^b=cg,f^b=fi,g^b=gi,e^c=eh,f^c=fhi,g^c=ghi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320528, #72320565, #72320806
maximal normal subgroup of	#72320528, #72320565, #72320806
maximal quotient of	#72322949

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406506);