#491231

order	512
nilpotency class	4
isoclinism	isoclinic class of #491227
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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 ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

463 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 463 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,6)(4,7)(5,9)(8,12)(10,14)(11,13)(15,16)(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);(1,3)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(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);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(19,30)(24,38)(28,41)(31,33)(32,45)(39,43)(42,47)(44,46);(3,10)(6,14)(11,16)(13,15)(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=i,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^b=dh,f^c=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398427);