#148371

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	#2092
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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=i,f^2=h,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, 55567);