#463214

order	512
nilpotency class	3
isoclinism	isoclinic class of #456400
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

932 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ei,c^2=efi,d^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^c=de,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, 370410);