#122139

order	512
nilpotency class	3
isoclinism	isoclinic class of #122137
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

544 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)(3,14,18,41,15,43,45,64)(4,10,22,39,23,36,52,62)(6,13,24,42,21,38,25,53)(8,27,30,16,29,47,56,19)(9,31,35,20,34,50,61,51)(17,37,48,63,44,33,60,57)(28,54,55,46,32,58,59,49);(2,8,11,29)(3,15)(4,20)(5,24)(9,32,34,28)(10,37,36,33)(12,41,40,64)(13,14,38,43)(16,19)(17,46)(18,47)(23,51)(25,26)(27,45)(30,53,56,42)(35,57,61,63)(39,55,62,59)(44,49)(48,60)(54,58);(2,9,11,34)(3,16)(4,20)(5,25)(8,28,29,32)(10,13,36,38)(12,39,40,62)(14,37,43,33)(15,19)(17,44)(22,52)(23,51)(24,26)(30,57,56,63)(31,50)(35,42,61,53)(41,59,64,55)(46,49)(48,54)(58,60)

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,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=gh,d^c=dhi,e^c=eh,g^c=ghi

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, 29335);