#489109

order	512
nilpotency class	4
isoclinism	isoclinic class of #488916
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	#61284	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

698 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396305);