#490346

order	512
nilpotency class	4
isoclinism	isoclinic class of #490107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33778	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

658 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=ghi,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dghi,e^a=eg,f^a=fh,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397542);