#490347

order	512
nilpotency class	4
isoclinism	isoclinic class of #490115
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
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

661 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)(20,23)(21,36)(22,40)(24,35)(25,31)(26,42)(27,38)(29,30)(32,48)(33,47)(34,39)(37,44)(41,46)(43,45);(1,2)(3,7)(4,12)(5,8)(6,11)(9,14)(10,15)(13,16)(17,19)(18,25)(20,29)(21,37)(22,41)(23,33)(24,43)(26,45)(27,46)(28,40)(30,39)(31,42)(32,38)(34,48)(35,44)(36,47);(1,3,5,10)(2,7,8,15)(4,13,11,9)(6,14,12,16)(17,20,23,18)(19,29,33,25)(21,34,22,26)(24,35,38,27)(28,30,47,31)(32,46,43,44)(36,42,40,39)(37,48,41,45);(2,8)(6,12)(7,15)(14,16)(17,21,38,42,23,22,24,39)(18,26,35,36,20,34,27,40)(19,30,43,41,33,31,32,37)(25,28,46,48,29,47,44,45)

Polycyclic presentation:

2^9|c^2=h,d^2=gh,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dgh,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

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397543);