#490358

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
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

683 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)(19,31)(20,35)(21,29)(22,28)(23,30)(24,32)(25,33)(26,37)(27,38)(34,41)(36,46)(39,45)(40,44)(42,48)(43,47);(1,2)(3,7)(4,12)(5,8)(6,11)(9,14)(10,15)(13,16)(17,19)(18,24)(20,36)(21,33)(22,40)(23,34)(25,26)(27,44)(28,42)(29,46)(30,43)(31,35)(32,38)(37,47)(39,48)(41,45);(1,3,5,10)(2,7,8,15)(4,13,11,9)(6,14,12,16)(17,19)(18,25)(20,32)(21,40)(22,33)(23,34)(24,26)(27,43)(28,41)(29,31)(30,44)(35,46)(36,38)(37,48)(39,47)(42,45);(2,8)(6,12)(7,15)(14,16)(17,20,21,37,23,39,22,38)(18,26,28,35,30,27,29,45)(19,32,40,48,34,47,33,36)(24,41,46,44,43,31,42,25)

Polycyclic presentation:

2^9|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, 397554);