#148370

order	512
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#469
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2034
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

668 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)(5,6,10,17,13,19,12,38)(7,20,23,52,25,53,35,41)(8,26,29,47,32,49,31,15)(9,18,11,36,34,50,33,16)(14,24,40,22,43,58,42,59)(21,54,56,44,57,68,60,51)(27,62,61,39,63,66,55,67)(28,64,65,46,37,45,30,48);(5,7,9,22,12,23,33,58,13,25,34,59,10,35,11,24)(6,14,16,20,17,42,18,41,19,43,36,53,38,40,50,52)(8,27,28,57,31,61,30,60,32,63,37,21,29,55,65,56)(15,44,45,39,26,54,48,62,47,51,64,67,49,68,46,66);(2,4)(5,8)(6,15)(7,21)(9,30)(10,31)(11,37)(12,29)(13,32)(14,39)(16,48)(17,49)(18,45)(19,47)(20,51)(22,27)(23,60)(24,61)(25,57)(26,38)(28,33)(34,65)(35,56)(36,46)(40,62)(41,54)(42,66)(43,67)(44,53)(50,64)(52,68)(55,58)(59,63)

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,e^b=ei,g^b=gi,d^c=dh,e^c=ei,f^c=fh,g^c=gi,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, 55566);