#493453

order	512
nilpotency class	4
isoclinism	isoclinic class of #493451
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1782
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3428 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2131 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)(3,15)(4,20,23,36)(5,11)(6,26,29,12)(7,13)(8,14)(9,18,33,51)(10,21,35,24)(16,44,48,63)(17,42)(19,43)(22,55)(25,39,28,37)(27,53)(30,58)(31,59)(32,49,40,57)(34,54)(38,56)(41,47,60,45)(46,61)(50,52)(62,64);(1,3)(2,9)(4,16)(5,17)(6,18)(7,30)(8,19)(10,31)(11,32)(12,15)(13,40)(14,33)(20,52)(21,45)(22,46)(23,48)(24,47)(25,49)(26,43)(27,50)(28,57)(29,51)(34,60)(35,62)(36,61)(37,42)(38,41)(39,58)(44,53)(54,59)(55,63)(56,64);(1,4)(2,10,14,35)(3,16)(5,24)(6,27)(7,21)(8,23)(9,31,33,62)(11,36,13,20)(12,38,26,34)(15,41,43,60)(17,47)(18,50)(19,48)(22,29)(25,54)(28,56)(30,45)(32,61,40,52)(37,53,39,55)(42,44,58,63)(46,51)(49,59)(57,64);(1,5,8,7)(2,11,14,13)(3,17,19,30)(4,21,23,24)(6,25,29,28)(9,32,33,40)(10,20,35,36)(12,37,26,39)(15,42,43,58)(16,45,48,47)(18,49,51,57)(22,54,27,56)(31,52,62,61)(34,53,38,55)(41,63,60,44)(46,59,50,64)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400649);