#243146

order	512
nilpotency class	3
isoclinism	isoclinic class of #243142
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
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

528 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150342);