#148285

order	512
nilpotency class	4
isoclinism	isoclinic class of #134701
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1809
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

651 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)(3,13,16,35)(4,18,20,9)(5,22)(6,25)(7,11)(8,15,28,40)(10,23)(12,26)(14,34)(17,43,42,36)(19,46,47,50)(21,32,49,45)(24,51)(27,54)(29,56,53,41)(30,37)(31,48,44,63)(33,52)(38,57)(39,62)(55,60)(58,64)(59,61);(1,3,5,15,6,17,23,41,7,16,24,40,26,42,52,56)(2,8,10,13,12,29,22,36,11,28,33,35,25,53,51,43)(4,14,19,37,21,39,48,60,20,38,47,54,49,61,63,64)(9,30,31,57,32,58,50,62,18,27,44,34,45,55,46,59);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,27)(10,31)(11,18)(12,32)(13,34)(15,37)(16,38)(17,39)(22,50)(23,48)(24,47)(25,45)(26,49)(28,30)(29,55)(33,44)(35,57)(36,59)(40,54)(41,60)(42,61)(43,62)(46,51)(52,63)(53,58)(56,64)

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,e^b=ei,g^b=gi,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, 55481);