#463339

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

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

807 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)(5,6,8,10)(7,9,11,12)(13,14,19,24)(15,25,30,44)(16,31,33,21)(17,22,35,41)(18,36,26,23)(20,37,38,29)(27,39,34,43)(28,42,32,40);(5,7)(6,9)(8,11)(10,12)(13,15,26,37,19,30,18,29)(14,20,36,44,24,38,23,25)(16,17,28,27,33,35,32,34)(21,39,42,22,31,43,40,41);(1,2,4,3)(7,11)(9,12)(13,16,19,33)(14,21,24,31)(15,27,30,34)(17,29,35,37)(18,28,26,32)(20,22,38,41)(23,42,36,40)(25,43,44,39);(1,3,4,2)(13,17,26,27,19,35,18,34)(14,22,36,43,24,41,23,39)(15,28,37,33,30,32,29,16)(20,31,44,40,38,21,25,42)

Polycyclic presentation:

2^9|a^2=eg,b^2=ei,c^2=f,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=ei,g^c=gi,h^c=hi,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, 370535);