#490636

order	512
nilpotency class	4
isoclinism	isoclinic class of #490474
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	C₂×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

450 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 142 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,16)(14,15)(18,23)(20,31)(21,32)(24,33)(25,34)(26,38)(27,37)(29,44)(30,43)(35,36)(39,42)(40,45)(41,48)(46,47);(2,6)(7,15)(8,13)(9,16)(17,18,24,38,21,26,20,23,22,27,31,36,32,35,33,37)(19,25,39,48,30,41,29,34,28,40,44,47,43,46,42,45);(1,3,4,10)(2,7,6,15)(5,14,12,11)(8,16,13,9)(17,19)(18,25)(20,29)(21,30)(22,28)(23,34)(24,39)(26,41)(27,40)(31,44)(32,43)(33,42)(35,46)(36,47)(37,45)(38,48);(19,28)(25,40)(29,42)(30,43)(34,45)(39,44)(41,46)(47,48)

Polycyclic presentation:

2^9|b^2=eghi,c^2=h,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^c=di,f^c=fh

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, 397832);