#243879

order	512
nilpotency class	3
isoclinism	isoclinic class of #232809
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

630 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=h,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,d^c=dh,e^c=ei,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151075);