#491272

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	#19332	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 423 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=eg,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, 398468);