#149401

order	512
nilpotency class	4
isoclinism	isoclinic class of #140094
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1919
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11378
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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=de,d^2=hi,e^2=hi,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=ghi,h^b=hi,d^c=dhi,e^c=eh,f^c=fh,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, 56597);