#148372

order	512
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#482
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

670 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)(5,6,10,17,13,19,12,39)(7,20,23,53,24,50,35,54)(8,25,28,47,30,49,29,15)(9,31,33,18,11,36,34,16)(14,40,42,22,44,60,43,37)(21,55,57,62,58,45,61,52)(26,41,63,66,56,68,59,67)(27,64,38,51,65,46,32,48);(5,7,9,22,12,23,34,60,13,24,11,37,10,35,33,40)(6,14,16,20,17,43,31,54,19,44,18,50,39,42,36,53)(8,26,27,61,29,63,32,21,30,56,65,57,28,59,38,58)(15,45,46,66,25,62,48,41,47,55,64,67,49,52,51,68);(2,4)(5,8)(6,15)(7,21)(9,32)(10,29)(11,38)(12,28)(13,30)(14,41)(16,48)(17,49)(18,51)(19,47)(20,52)(22,59)(23,61)(24,58)(25,39)(26,40)(27,34)(31,46)(33,65)(35,57)(36,64)(37,63)(42,67)(43,66)(44,68)(45,53)(50,62)(54,55)(56,60)

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=i,f^2=h,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,f^b=fi,g^b=gi,d^c=dh,e^c=ei,f^c=fhi,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, 55568);