#396126

order	512
nilpotency class	3
isoclinism	isoclinic class of #396125
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14730
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

1338 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:

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

Polycyclic presentation:

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

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, 303322);