#490359

order	512
nilpotency class	4
isoclinism	isoclinic class of #490121
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	#33785	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

655 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)(19,29)(20,23)(21,36)(22,26)(24,37)(25,32)(27,39)(28,40)(30,38)(31,46)(33,43)(34,47)(35,41)(42,45)(44,48);(1,2)(3,7)(4,12)(5,8)(6,11)(9,14)(10,15)(13,16)(17,19,23,34)(18,25,20,30)(21,38,39,32)(22,31,41,48)(24,42,40,33)(26,29,35,47)(27,45,36,43)(28,46,37,44);(1,3,5,10)(2,7,8,15)(4,13,11,9)(6,14,12,16)(17,20,23,18)(19,30,34,25)(21,35,39,26)(22,27,41,36)(24,37,40,28)(29,38,47,32)(31,45,48,43)(33,46,42,44);(2,8)(6,12)(7,15)(14,16)(17,21,24,22,23,39,40,41)(18,26,28,36,20,35,37,27)(19,31,33,38,34,48,42,32)(25,43,44,29,30,45,46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397555);