#493423

order	512
nilpotency class	4
isoclinism	isoclinic class of #493420
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1781
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#62859	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#585 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

3249 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)(3,15,19,38)(4,20,23,40)(5,11)(6,26,29,12)(7,13)(8,14)(9,18)(10,21,35,30)(16,43)(17,42,24,55)(22,52)(25,39,28,37)(27,33)(31,59,61,64)(32,48)(34,51)(36,56)(41,44)(45,49,53,60)(46,62)(47,63)(50,54)(57,58);(1,3,8,19)(2,9,14,33)(4,16,23,47)(5,24,7,17)(6,27,29,18)(10,31,35,61)(11,36,13,32)(12,38,26,15)(20,49,40,60)(21,46,30,44)(22,53,54,45)(25,48,28,56)(34,62,57,41)(37,42,39,55)(43,50,63,52)(51,64,58,59);(1,4)(2,10)(3,16)(5,21)(6,22)(7,30)(8,23)(9,31)(11,20)(12,34)(13,40)(14,35)(15,41)(17,44)(18,45)(19,47)(24,46)(25,51)(26,57)(27,53)(28,58)(29,54)(32,60)(33,61)(36,49)(37,50)(38,62)(39,52)(42,63)(43,55)(48,64)(56,59);(1,5,8,7)(2,11,14,13)(3,17,19,24)(4,21,23,30)(6,25,29,28)(9,32,33,36)(10,20,35,40)(12,37,26,39)(15,42,38,55)(16,44,47,46)(18,48,27,56)(22,51,54,58)(31,60,61,49)(34,50,57,52)(41,63,62,43)(45,64,53,59)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303715, #72303736, #72303754, …
maximal normal subgroup of	#72303715, #72303736, #72303754, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400619);