#148436

order	512
nilpotency class	4
isoclinism	isoclinic class of #140776
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1970
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11123
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

665 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)(3,15)(4,21,26,43)(5,28)(6,12)(7,33)(8,14)(9,18)(10,24,41,27)(11,16)(13,30)(17,50,53,39)(19,47)(20,36)(22,34,51,63)(23,59,25,40)(29,42,58,45)(31,44)(32,48)(35,49)(37,61)(38,55)(46,52,57,56)(54,62,60,64);(1,3,16,49,7,19,31,55,8,20,30,61,6,32,5,18)(2,9,28,48,12,37,13,36,14,38,44,47,33,35,11,15)(4,22,58,53,27,54,25,56,26,51,29,17,24,60,23,52)(10,39,45,34,21,57,59,64,41,50,42,63,43,46,40,62);(1,4)(2,10)(3,17)(5,29)(6,27)(7,24)(8,26)(9,34)(11,42)(12,43)(13,45)(14,41)(15,46)(16,25)(18,54)(19,56)(20,53)(21,33)(22,49)(23,30)(28,59)(31,58)(32,52)(35,64)(36,57)(37,62)(38,63)(39,47)(40,44)(48,50)(51,61)(55,60)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=hi,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,e^b=ei,d^c=dhi,e^c=ei,f^c=fh,g^c=gi,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, 55632);