#148971

order	512
nilpotency class	4
isoclinism	isoclinic class of #140148
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1926
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11257
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

435 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)(3,13,17,8)(4,9)(5,10)(6,24)(7,12)(11,23)(14,30)(15,31,46,43)(16,32,49,44)(18,41,28,33)(19,51,52,34)(20,53)(21,37)(22,35)(25,38,56,58)(26,59)(27,60)(29,42)(36,54)(39,57)(40,45)(47,61,50,63)(48,64,55,62);(1,3,14,18,5,17,7,28)(2,8,30,33,10,13,12,41)(4,19,21,55,22,52,29,48)(6,25,45,49,23,56,27,16)(9,34,37,64,35,51,42,62)(11,38,60,44,24,58,40,32)(15,26,50,54,46,57,47,20)(31,39,63,53,43,59,61,36);(1,4)(2,9,10,35)(3,15)(5,22)(6,26)(7,29)(8,31,13,43)(11,39,24,59)(12,42,30,37)(14,21)(16,48)(17,46)(18,50)(19,25)(20,27)(23,57)(28,47)(32,62,44,64)(33,63,41,61)(34,58,51,38)(36,60,53,40)(45,54)(49,55)(52,56)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,e^b=ei,f^b=fh,g^b=gi,h^b=hi,e^c=ei,f^c=fh,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, 56167);