#122145

order	512
nilpotency class	3
isoclinism	isoclinic class of #122142
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1446
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=gh,d^c=dhi,e^c=eh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29341);