#243026

order	512
nilpotency class	3
isoclinism	isoclinic class of #10637
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#397
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#397
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

647 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150222);