#243842

order	512
nilpotency class	3
isoclinism	isoclinic class of #232749
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

615 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:

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

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,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,d^c=dh,e^c=ei,f^c=fi

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, 151038);