#400818

order	512
nilpotency class	3
isoclinism	isoclinic class of #393928
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14807
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1246 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308014);