#423849

order	512
nilpotency class	3
isoclinism	isoclinic class of #423848
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15316
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1261 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:

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

Polycyclic presentation:

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

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, 331045);