#436715

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
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	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1195 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)(5,6,8,10)(7,11,12,9)(13,14,19,25,18,26,20,24)(15,27,30,22,16,21,29,37)(17,32,34,39,35,40,33,23)(28,42,43,41,31,38,44,36);(5,7,8,12)(6,9,10,11)(13,15,19,29,18,16,20,30)(14,21,24,37,26,27,25,22)(17,31,33,44,35,28,34,43)(23,36,32,38,39,41,40,42);(6,10)(9,11)(13,16,18,15)(14,22,26,37)(17,31,35,28)(19,29,20,30)(21,24,27,25)(23,38,39,42)(32,36,40,41)(33,43,34,44);(2,4)(7,12)(9,11)(13,17,19,34,18,35,20,33)(14,23,25,32,26,39,24,40)(15,28,30,43,16,31,29,44)(21,36,37,42,27,41,22,38)

Polycyclic presentation:

2^9|a^2=eg,b^2=f,c^2=i,d^2=efi,f^2=i,g^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,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=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, 343911);