#491545

order	512
nilpotency class	4
isoclinism	isoclinic class of #491511
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

557 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 557 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398741);