#466815

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	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1088 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=efi,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,g^c=gi,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, 374011);