#489128

order	512
nilpotency class	4
isoclinism	isoclinic class of #488889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂³	#447
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

669 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 669 conjugacy classes of subgroups — too many to draw. Only its 156 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396324);