#456097

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1099 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,27)(17,26)(19,23)(20,24)(21,30)(22,34)(25,31)(28,41)(29,37)(32,38)(33,40)(35,39)(36,44)(42,43);(6,9)(10,12)(13,15,19,24,20,23,18,14)(16,25,30,21,31,27,29,37)(17,26,34,40,35,39,33,22)(28,38,43,36,44,42,32,41);(7,11)(10,12)(13,16,20,31)(14,21,24,37)(15,25,23,27)(17,32,35,43)(18,30,19,29)(22,38,40,42)(26,41,39,36)(28,34,44,33);(2,4)(5,7,8,11)(6,10,9,12)(13,17,19,34,20,35,18,33)(14,22,15,26,24,40,23,39)(16,28,30,43,31,44,29,32)(21,36,27,42,37,41,25,38)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363293);