#464157

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1256 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)(5,6,8,9)(7,10,11,12)(13,14)(15,21)(16,31)(17,33)(18,24)(19,23)(20,25)(22,35)(26,27)(28,40)(29,39)(30,41)(32,43)(34,36)(37,38)(42,44);(6,9)(10,12)(13,15,19,29,20,30,18,28)(14,21,23,39,25,41,24,40)(16,32,34,42,22,38,33,26)(17,27,31,43,36,44,35,37);(2,4)(5,7)(6,10)(8,11)(9,12)(13,16,20,22)(14,17,25,36)(15,26,30,42)(18,34,19,33)(21,37,41,43)(23,35,24,31)(27,39,44,40)(28,32,29,38);(7,11)(10,12)(13,17)(14,22)(15,27)(16,25)(18,35)(19,31)(20,36)(21,38)(23,33)(24,34)(26,39)(28,37)(29,43)(30,44)(32,41)(40,42)

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^c=de,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=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, 371353);