#473745

order	512
nilpotency class	3
isoclinism	isoclinic class of #457092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

781 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,5)(3,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,18,22,28)(19,27,32,25)(20,34,36,45)(21,39,33,42)(23,30,24,29)(26,38,44,37)(31,47,48,43)(35,41,40,46);(2,5)(6,7)(9,11,13,12)(10,14,16,15)(17,19,24,33,22,32,23,21)(18,25,29,39,28,27,30,42)(20,35,37,31,36,40,38,48)(26,43,45,41,44,47,34,46);(1,3,4,8)(2,6,5,7)(9,11,13,12)(10,15,16,14)(17,20,23,37,22,36,24,38)(18,26,29,34,28,44,30,45)(19,31,21,40,32,48,33,35)(25,41,39,43,27,46,42,47);(3,8)(6,7)(9,12,13,11)(10,15,16,14)(17,21,23,32,22,33,24,19)(18,27,29,42,28,25,30,39)(20,35,37,31,36,40,38,48)(26,41,34,43,44,46,45,47)

Polycyclic presentation:

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

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, 380941);