#148880

order	512
nilpotency class	4
isoclinism	isoclinic class of #134701
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	#11231
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

465 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)(3,15,19,45)(4,21)(5,12)(6,28)(7,32)(8,13)(9,17,35,50)(10,23)(11,26)(14,31)(16,37,48,33)(18,36,29,58)(20,34,52,41)(22,40)(24,57)(25,42)(27,43)(30,39)(38,56)(44,51,61,47)(46,60,63,53)(49,59,64,62)(54,55);(1,3,5,17,7,20,26,18,8,19,27,50,31,52,6,29)(2,9,11,15,14,36,12,41,13,35,28,45,32,58,43,34)(4,16,22,46,24,49,23,51,25,48,55,63,56,64,30,47)(10,37,38,61,40,62,42,60,39,33,57,44,54,59,21,53);(1,4,5,22,7,24,26,23,8,25,27,55,31,56,6,30)(2,10,11,38,14,40,12,42,13,39,28,57,32,54,43,21)(3,16,17,46,20,49,18,51,19,48,50,63,52,64,29,47)(9,33,15,44,36,59,41,53,35,37,45,61,58,62,34,60)

Polycyclic presentation:

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

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, 56076);