#243875

order	512
nilpotency class	3
isoclinism	isoclinic class of #232789
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

658 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)(3,11,13,16,14,15,12,8)(17,18,22,27,23,28,21,26)(19,29,31,38,32,43,36,24)(20,25,33,40,34,41,35,42)(30,45,47,37,48,44,46,39);(2,7)(4,5)(8,11)(9,10)(12,13)(15,16)(17,19,21,31,23,32,22,36)(18,24,26,29,28,38,27,43)(20,30,35,47,34,48,33,46)(25,39,42,45,41,37,40,44);(1,3,6,14)(2,8,10,16)(4,13,5,12)(7,11,9,15)(17,20,23,34)(18,25,28,41)(19,30,32,48)(21,35,22,33)(24,37,38,39)(26,42,27,40)(29,44,43,45)(31,47,36,46);(1,4,6,5)(2,7,10,9)(3,12,14,13)(8,15,16,11)(17,21,23,22)(18,26,28,27)(19,31,32,36)(20,33,34,35)(24,29,38,43)(25,40,41,42)(30,46,48,47)(37,45,39,44)

Polycyclic presentation:

2^9|a^2=e,b^2=efh,c^2=h,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151071);