#243849

order	512
nilpotency class	3
isoclinism	isoclinic class of #232771
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

669 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)(19,29,32,39,33,38,31,24)(20,25,34,41,35,42,36,43)(30,45,46,37,47,44,48,40);(2,7)(4,5)(8,11)(9,10)(12,13)(15,16)(17,19,21,32,23,33,22,31)(18,24,26,29,28,39,27,38)(20,30,36,46,35,47,34,48)(25,40,43,45,42,37,41,44);(1,3,6,14)(2,8,10,16)(4,13,5,12)(7,11,9,15)(17,20,23,35)(18,25,28,42)(19,30,33,47)(21,36,22,34)(24,37,39,40)(26,43,27,41)(29,44,38,45)(31,48,32,46);(1,4,6,5)(2,7,10,9)(3,12,14,13)(8,15,16,11)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,34,35,36)(24,38,39,29)(25,41,42,43)(30,46,47,48)(37,44,40,45)

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=dh,e^b=eh,f^b=fh,d^c=dh,e^c=ei,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151045);