#464753

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1518 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)(6,9)(10,12)(13,14,18,21,19,24,17,23)(15,20,27,35,28,36,25,34)(16,29,31,38,32,37,30,22)(26,40,39,44,42,43,41,33);(5,6,8,9)(7,10,11,12)(13,15,18,27,19,28,17,25)(14,20,21,35,24,36,23,34)(16,26,30,41,32,42,31,39)(22,33,37,43,38,44,29,40);(2,4)(7,11)(10,12)(14,21)(15,25)(17,18)(22,37)(23,24)(26,39)(27,28)(29,38)(30,31)(34,35)(40,43)(41,42);(5,7,8,11)(6,10,9,12)(13,16)(14,22)(15,26)(17,31)(18,30)(19,32)(20,33)(21,37)(23,29)(24,38)(25,39)(27,41)(28,42)(34,40)(35,43)(36,44)

Polycyclic presentation:

2^9|a^2=fi,b^2=eg,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,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, 371949);