#500165

order	512
nilpotency class	4
isoclinism	isoclinic class of #500164
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	#61054	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

949 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:

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

Polycyclic presentation:

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

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307076, #72307079, #72307097, …
maximal normal subgroup of	#72307076, #72307079, #72307097, …
maximal quotient of	C₂×#500165

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407361);