#491241

order	512
nilpotency class	4
isoclinism	isoclinic class of #491239
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
prime radical	#19187	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

466 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 128 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)(18,23)(20,31)(21,32)(24,33)(25,34)(26,35)(27,38)(29,44)(30,43)(36,37)(39,42)(40,48)(41,47)(45,46);(1,3)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(17,18,24,38,21,27,20,23,22,26,31,37,32,36,33,35)(19,25,39,48,30,40,29,34,28,41,44,46,43,45,42,47);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,19)(18,25)(20,29)(21,30)(22,28)(23,34)(24,39)(26,41)(27,40)(31,44)(32,43)(33,42)(35,47)(36,45)(37,46)(38,48);(3,10)(6,14)(11,16)(13,15)(18,26)(19,28)(23,35)(27,36)(29,42)(30,43)(37,38)(39,44)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398437);