#148864

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×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

479 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=dh,e^2=hi,g^2=h,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, 56060);