#148965

order	512
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	#2085
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

436 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,e^b=ei,f^b=fh,g^b=gi,h^b=hi,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, 56161);