#412870

order	512
nilpotency class	3
isoclinism	isoclinic class of #402446
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

966 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(5,6,12,18)(7,19,23,42)(8,24,27,14)(9,29,32,40)(10,17,11,16)(13,33,36,22)(15,30,41,31)(20,45,48,43)(21,49,51,35)(25,54,56,59)(26,38,28,39)(34,47,52,58)(37,57,53,55)(44,60,62,50)(46,61,66,67)(63,65,64,68);(1,2)(3,4)(5,7)(6,13)(8,20,27,48)(9,21)(10,33)(11,22)(12,23)(14,34,24,52)(15,35)(16,42)(17,19)(18,36)(25,46,56,66)(26,58,28,47)(29,44)(30,60)(31,50)(32,51)(37,61,53,67)(38,45,39,43)(40,62)(41,49)(54,64,59,63)(55,68,57,65);(2,4)(5,8,10,26,12,27,11,28)(6,14,16,38,18,24,17,39)(7,20,22,47,23,48,33,58)(9,25,30,55,32,56,31,57)(13,34,19,43,36,52,42,45)(15,37,40,54,41,53,29,59)(21,46,50,65,51,66,60,68)(35,61,44,63,49,67,62,64);(5,9,10,30,12,32,11,31)(6,15,16,40,18,41,17,29)(7,21,33,60,23,51,22,50)(8,25,26,55,27,56,28,57)(13,35,42,62,36,49,19,44)(14,37,38,54,24,53,39,59)(20,46,58,68,48,66,47,65)(34,61,45,64,52,67,43,63)

Polycyclic presentation:

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

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, 320066);