#242386

order	512
nilpotency class	3
isoclinism	isoclinic class of #242377
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#396
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

581 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)(3,5)(4,12)(6,9)(7,10)(8,13)(11,15)(14,16)(17,18,23,26,22,27,21,35)(19,28,37,39,30,45,31,24)(20,25,36,41,34,42,33,43)(29,44,47,38,46,40,32,48);(1,3,6,10,7,9,5,2)(4,11,13,8,15,12,14,16)(17,19,23,37,22,30,21,31)(18,24,35,45,27,39,26,28)(20,32,36,29,34,47,33,46)(25,40,43,38,42,44,41,48);(1,4,7,15)(2,8,10,16)(3,11,9,12)(5,13,6,14)(17,20,22,34)(18,25,27,42)(19,29,30,46)(21,36,23,33)(24,38,39,48)(26,43,35,41)(28,44,45,40)(31,47,37,32);(1,5,7,6)(2,3,10,9)(4,13,15,14)(8,11,16,12)(19,30)(24,39)(28,45)(29,46)(31,37)(32,47)(38,48)(40,44)

Polycyclic presentation:

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

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, 149582);