#243049

order	512
nilpotency class	3
isoclinism	isoclinic class of #243042
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	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

588 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:

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

Polycyclic presentation:

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

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, 150245);