#148865

order	512
nilpotency class	4
isoclinism	isoclinic class of #137665
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1895
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11228
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

471 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=dh,e^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,f^b=fi,d^c=di,e^c=eh,g^c=ghi,h^c=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, 56061);