#243001

order	512
nilpotency class	3
isoclinism	isoclinic class of #242992
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	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

584 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=e,d^2=hi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^b=gi,f^c=fh,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, 150197);