#500159

order	512
nilpotency class	4
isoclinism	isoclinic class of #500158
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24033
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61064	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2930 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

941 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)(3,8)(4,18)(5,23)(6,29)(7,13)(9,21)(10,27)(11,43)(12,26)(14,49)(15,41)(16,47)(17,36)(19,32)(20,39)(22,42)(24,33)(25,40)(28,46)(30,35)(31,45)(34,60)(37,51)(38,50)(44,54)(48,59)(52,62)(53,61)(55,63)(56,57)(58,64);(1,3,7,17)(2,8,13,36)(4,19,20,57)(5,24,28,60)(6,30,26,16)(9,37,38,49)(10,41,11,44)(12,47,29,35)(14,50,51,21)(15,43,54,27)(18,56,39,32)(22,53,31,58)(23,33,46,34)(25,52,59,55)(40,63,48,62)(42,64,45,61);(1,4,6,31,7,20,26,22)(2,9,12,48,13,38,29,40)(3,14,16,55,17,51,30,52)(5,25,27,50,28,59,43,21)(8,32,35,64,36,56,47,61)(10,42,23,39,11,45,46,18)(15,53,24,57,54,58,60,19)(33,62,41,37,34,63,44,49);(1,2)(3,8)(4,9)(5,11)(6,12)(7,13)(10,28)(14,32)(15,34)(16,35)(17,36)(18,50)(19,37)(20,38)(21,39)(22,40)(23,43)(24,44)(25,45)(26,29)(27,46)(30,47)(31,48)(33,54)(41,60)(42,59)(49,57)(51,56)(52,61)(53,63)(55,64)(58,62)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307714, #72307725, #72307943, …
maximal normal subgroup of	#72307714, #72307725, #72307943, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407355);