#490670

order	512
nilpotency class	4
isoclinism	isoclinic class of #490474
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
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

430 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(18,23)(20,24)(21,32)(25,34)(26,42)(27,37)(28,43)(30,38)(31,46)(33,39)(35,36)(40,45)(41,48)(44,47);(17,18,24,36,21,26,33,37,22,27,39,42,32,35,20,23)(19,28,38,48,31,44,45,34,29,25,40,47,46,41,30,43);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19)(18,25)(20,30)(21,31)(22,29)(23,34)(24,38)(26,41)(27,28)(32,46)(33,45)(35,44)(36,47)(37,43)(39,40)(42,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(19,29)(25,28)(30,45)(31,46)(34,43)(38,40)(41,44)(47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397866);