#489134

order	512
nilpotency class	4
isoclinism	isoclinic class of #488901
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 ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 659 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)(18,23)(19,27)(20,30)(21,33)(25,39)(26,43)(28,46)(29,45)(31,35)(32,36)(34,37)(38,41)(42,44)(47,48);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,18,22,25)(19,24,29,40)(20,31,34,43)(21,35,33,26)(23,37,39,30)(27,38,45,47)(28,44,36,42)(32,41,46,48);(17,19,22,29)(18,24,25,40)(20,32,34,46)(21,36,33,28)(23,38,39,47)(26,44,35,42)(27,37,45,30)(31,41,43,48);(18,25)(23,39)(24,40)(26,35)(31,43)(38,47)(41,48)(42,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,h^a=hi,d^b=di,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, 396330);