#148647

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	#2076
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

636 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)(7,18,22,41)(8,24,28,58)(9,14)(10,32)(11,16)(12,20,37,23)(13,27,39,62)(15,30)(17,31)(19,47)(21,52,54,36)(25,59)(26,57,29,43)(33,40,63,38)(34,42)(35,53)(44,61)(45,60)(46,51,67,55)(48,56)(49,65)(50,68)(64,66);(1,2)(3,4)(5,7)(6,12)(8,25)(9,23)(10,21)(11,22)(13,35)(14,41)(15,36)(16,37)(17,18)(19,28)(20,31)(24,56)(26,48)(27,60)(29,50)(30,55)(32,46)(33,64)(34,54)(38,45)(39,66)(40,65)(42,67)(43,59)(44,52)(47,57)(49,62)(51,61)(53,63)(58,68);(2,4)(5,8,11,28)(6,13)(7,19)(9,29,31,26)(10,33,34,63)(12,35,37,66)(14,40)(15,43)(16,39)(17,38)(18,45,41,65)(20,50)(21,53)(22,25)(23,48)(24,32)(27,61,62,30)(36,59,52,47)(42,58)(44,57)(46,56,67,68)(49,55)(51,60)(54,64)

Polycyclic presentation:

2^9|a^2=g,c^2=dh,d^2=i,e^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=di,g^b=gi,d^c=di,e^c=eh,f^c=fi,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, 55843);