#148335

order	512
nilpotency class	4
isoclinism	isoclinic class of #135850
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	#11096
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

645 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=cf,f^b=fi,f^c=fi,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, 55531);