#243144

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

529 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=e,d^2=h,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, 150340);