#146784

order	512
nilpotency class	4
isoclinism	isoclinic class of #12826
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1804
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

942 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=hi,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,h^a=hi,d^b=dhi,e^b=eh,h^b=hi,d^c=dh,e^c=ehi,f^c=fi,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, 53980);