#490665

order	512
nilpotency class	4
isoclinism	isoclinic class of #12433
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×#466
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

436 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)(17,18)(19,28)(20,25)(21,35)(22,37)(23,27)(24,32)(26,29)(30,41)(31,42)(33,47)(34,40)(36,48)(38,45)(39,46)(43,44);(17,19,29,44,22,33,36,45,23,30,34,46,20,31,21,32)(18,24,35,42,25,39,40,41,27,38,48,47,37,43,26,28);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(19,30)(24,38)(28,41)(31,33)(32,45)(39,43)(42,47)(44,46);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,23,22)(18,25,27,37)(19,31,30,33)(21,34,36,29)(24,39,38,43)(26,35,40,48)(28,42,41,47)(32,46,45,44)

Polycyclic presentation:

2^9|b^2=eg,d^2=hi,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,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, 397861);