#490102

order	512
nilpotency class	4
isoclinism	isoclinic class of #490099
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 ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

704 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:

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

Polycyclic presentation:

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

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397298);