#148500

order	512
nilpotency class	4
isoclinism	isoclinic class of #13442
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

638 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)(7,19)(8,25)(9,32)(10,15)(11,36)(12,18)(13,21)(14,28)(16,34)(17,33)(20,52)(22,50)(23,39)(24,49)(26,61)(27,46)(29,45)(30,65)(31,43)(35,48)(37,47)(38,56)(40,59)(41,58)(42,62)(44,63)(51,55)(53,60)(54,66)(57,68)(64,67);(5,7,12,24)(6,13,18,41)(8,26,31,62)(9,22,34,59)(10,21,35,58)(11,37,33,23)(14,42,46,61)(15,47,48,39)(16,49,32,19)(17,50,36,40)(20,53,57,29)(25,52,43,68)(27,56,28,64)(30,54,63,55)(38,65,67,44)(45,66,60,51);(5,8,9,27,33,63,35,53,12,31,34,28,11,30,10,29)(6,14,15,43,36,60,16,44,18,46,48,25,17,45,32,65)(7,20,21,54,37,64,59,62,24,57,58,55,23,56,22,26)(13,38,19,51,50,68,39,42,41,67,49,66,40,52,47,61)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,d^2=hi,e^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=dh,e^b=eh,f^b=fhi,h^b=hi,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, 55696);