#490926

order	512
nilpotency class	4
isoclinism	isoclinic class of #490873
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

460 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 138 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)(3,6)(4,7)(5,13)(8,12)(9,11)(10,14)(15,16)(17,18,22,26)(19,27,31,41)(20,24,32,38)(21,34,28,25)(23,35,37,30)(29,40,43,48)(33,45,42,39)(36,46,47,44);(2,6)(7,14)(8,15)(9,13)(17,19,28,35,22,31,21,30)(18,23,34,41,26,37,25,27)(20,29,42,46,32,43,33,44)(24,36,45,48,38,47,39,40);(2,7)(4,12)(5,11)(6,14)(8,9)(10,16)(13,15)(17,20)(18,24)(19,29)(21,33)(22,32)(23,36)(25,39)(26,38)(27,40)(28,42)(30,44)(31,43)(34,45)(35,46)(37,47)(41,48);(20,32)(24,38)(29,43)(33,42)(36,47)(39,45)(40,48)(44,46)

Polycyclic presentation:

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

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, 398122);