#149327

order	512
nilpotency class	4
isoclinism	isoclinic class of #129558
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#418
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1824
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

480 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(5,6)(7,18,22,40)(8,24)(9,28)(10,32)(11,36)(12,17)(13,21,41,59)(14,25)(15,31)(16,19)(20,53,56,39)(23,46,37,52)(26,47)(27,43)(29,49,33,42)(30,50)(34,44)(35,48)(38,61)(45,63)(51,54,66,55)(57,65,60,67)(58,68,62,64);(1,2)(3,4)(5,7,19,23,12,22,11,37)(6,13,16,49,17,41,36,42)(8,20,26,60,27,56,38,57)(9,29,35,21,10,33,31,59)(14,39,50,65,45,53,44,67)(15,46,28,40,48,52,32,18)(24,51,47,68,43,66,61,64)(25,62,30,55,63,58,34,54);(2,4)(5,8,19,26,12,27,11,38)(6,14,16,50,17,45,36,44)(7,20,23,60,22,56,37,57)(9,30,35,63,10,34,31,25)(13,39,49,65,41,53,42,67)(15,47,28,43,48,61,32,24)(18,51,46,68,40,66,52,64)(21,58,33,54,59,62,29,55)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=h,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,g^b=gi,h^b=hi,d^c=dh,e^c=ehi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56523);