#400114

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	#14796
prime radical	#23434	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1389 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)(3,15,18,32,19,33,28,9)(4,10,22,36,24,37,23,40)(5,25,26,11,27,38,29,39)(16,42,44,30,45,34,20,50)(17,46,47,57,48,31,49,41)(21,52,53,35,55,59,54,60)(43,56,62,58,63,64,51,61);(1,3)(2,9)(4,20)(5,17)(6,28)(7,18)(8,19)(10,34)(11,31)(12,33)(13,15)(14,32)(16,23)(21,51)(22,45)(24,44)(25,41)(26,49)(27,48)(29,47)(30,36)(35,58)(37,42)(38,57)(39,46)(40,50)(43,54)(52,64)(53,63)(55,62)(56,59)(60,61);(1,4)(2,10)(3,16)(5,21)(6,22)(7,23)(8,24)(9,30)(11,35)(12,36)(13,40)(14,37)(15,34)(17,43)(18,44)(19,45)(20,28)(25,52)(26,53)(27,55)(29,54)(31,56)(32,50)(33,42)(38,59)(39,60)(41,58)(46,64)(47,62)(48,63)(49,51)(57,61);(1,5,6,26,8,27,7,29)(2,11,12,38,14,39,13,25)(3,17,28,49,19,48,18,47)(4,21,22,53,24,55,23,54)(9,31,33,57,32,46,15,41)(10,35,36,59,37,60,40,52)(16,43,20,51,45,63,44,62)(30,56,42,61,50,64,34,58)

Polycyclic presentation:

2^9|a^2=e,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi,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, 307310);