#497431

order	512
nilpotency class	4
isoclinism	isoclinic class of #497430
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#22436
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#62637	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1765 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

937 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)(3,14,19,42)(4,20,23,39)(5,24)(6,12)(7,31)(8,13)(9,17,35,51)(10,22,38,30)(11,28)(15,44)(16,32,43,18)(21,37,56,61)(25,34,50,36)(26,29)(27,40)(33,54)(41,48)(45,49)(46,60)(47,62)(52,64)(53,58,57,55)(59,63);(1,3)(2,9)(4,15)(5,25)(6,18)(7,32)(8,19)(10,33)(11,16)(12,34)(13,35)(14,26)(17,27)(20,52)(21,54)(22,47)(23,49)(24,42)(28,51)(29,50)(30,48)(31,36)(37,45)(38,59)(39,60)(40,43)(41,53)(44,61)(46,58)(55,64)(56,63)(57,62);(1,4,8,23)(2,10,13,38)(3,15,19,49)(5,21,29,56)(6,30,7,22)(9,33,35,59)(11,37,40,61)(12,39,31,20)(14,41,42,62)(16,45,43,44)(17,46,51,64)(18,48,32,47)(24,57,26,53)(25,54,50,63)(27,58,28,55)(34,60,36,52);(1,2)(3,16)(4,10)(5,26)(6,12)(7,31)(8,13)(9,25)(11,27)(14,32)(15,45)(17,34)(18,42)(19,43)(20,22)(21,53)(23,38)(24,29)(28,40)(30,39)(33,54)(35,50)(36,51)(37,58)(41,47)(44,49)(46,60)(48,62)(52,64)(55,61)(56,57)(59,63)

Polycyclic presentation:

2^9|a^2=g,c^2=i,e^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=de,e^b=ehi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,e^d=ehi,h^d=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285481, #72285570, #72286016
maximal normal subgroup of	#72285481, #72285570, #72286016
maximal quotient of	#72287358, #72287708, #72309426, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404627);