#148553

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	#2073
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

648 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=h,e^2=h,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fh,g^b=gi,h^b=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, 55749);