#493479

order	512
nilpotency class	4
isoclinism	isoclinic class of #493477
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#20585
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#62871	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #3485 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2159 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)(3,13)(4,9)(5,10)(6,24,25,11)(7,12)(8,16,28,43)(14,36)(15,37)(17,38)(18,44)(19,30)(20,49,50,31)(21,32)(22,51,52,34)(23,35)(26,40,29,46)(27,41,55,62)(33,48)(39,58)(42,60)(45,59)(47,57,64,56)(53,61,54,63);(1,3,7,17)(2,8,12,28)(4,18,21,14)(5,15,23,42)(6,16,25,43)(9,29,32,26)(10,27,35,55)(11,13,24,38)(19,45,48,39)(20,46,50,40)(22,41,52,62)(30,54,33,53)(31,44,49,36)(34,37,51,60)(47,63,64,61)(56,59,57,58);(1,4,7,21)(2,9,12,32)(3,14,17,18)(5,19,23,48)(6,20,25,50)(8,26,28,29)(10,33,35,30)(11,31,24,49)(13,36,38,44)(15,39,42,45)(16,40,43,46)(22,47,52,64)(27,54,55,53)(34,57,51,56)(37,59,60,58)(41,61,62,63);(1,5)(2,10)(3,15)(4,19)(6,22)(7,23)(8,27)(9,30)(11,34)(12,35)(13,37)(14,39)(16,41)(17,42)(18,45)(20,47)(21,48)(24,51)(25,52)(26,53)(28,55)(29,54)(31,56)(32,33)(36,58)(38,60)(40,61)(43,62)(44,59)(46,63)(49,57)(50,64)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400675);