#442352

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
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

921 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(2,6)(7,8)(10,13)(11,15)(17,18,23,29,24,30,22,28)(19,27,21,25,33,39,32,40)(20,26,36,44,37,45,35,43)(31,42,34,38,47,41,46,48);(1,2,5,6)(3,7,4,8)(9,10,12,13)(11,14,15,16)(17,19)(18,25)(20,34)(21,22)(23,32)(24,33)(26,41)(27,29)(28,39)(30,40)(31,36)(35,47)(37,46)(38,44)(42,45)(43,48);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,20,22,35,24,37,23,36)(18,26,28,43,30,45,29,44)(19,31,32,46,33,47,21,34)(25,38,27,42,40,48,39,41);(1,4,5,3)(2,7,6,8)(10,13)(14,16)(17,21,23,33,24,32,22,19)(18,27,29,25,30,39,28,40)(20,34,36,47,37,46,35,31)(26,42,44,38,45,41,43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 349548);