#148308

order	512
nilpotency class	4
isoclinism	isoclinic class of #135851
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1854
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11095
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

649 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)(3,13,16,37)(4,18,21,9)(5,23)(6,27)(7,11)(8,15,30,43)(10,25)(12,28)(14,39)(17,46,45,38)(19,36)(20,50,24,54)(22,33,52,48)(26,53)(29,42)(31,59,57,44)(32,51,34,55)(35,56)(40,47)(41,61)(49,62)(58,63)(60,64);(1,3,5,15,6,17,25,44,7,16,26,43,28,45,56,59)(2,8,10,13,12,31,23,38,11,30,35,37,27,57,53,46)(4,19,20,42,22,49,51,64,21,14,24,40,52,41,55,63)(9,29,32,39,33,58,54,62,18,47,34,36,48,60,50,61);(1,4)(2,9)(3,14)(5,24)(6,22)(7,21)(8,29)(10,34)(11,18)(12,33)(13,36)(15,42)(16,19)(17,41)(20,26)(23,50)(25,55)(27,48)(28,52)(30,47)(31,58)(32,35)(37,39)(38,61)(40,43)(44,64)(45,49)(46,62)(51,56)(53,54)(57,60)(59,63)

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,f^b=fi,g^b=gi,d^c=di,g^c=gi

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, 55504);