#489065

order	512
nilpotency class	4
isoclinism	isoclinic class of #488920
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

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

normal subgroups only

This group has 684 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) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

(2,5)(3,7)(6,8)(9,10,16,22)(11,23,25,56)(12,29,33,67)(13,20,36,52)(14,19,37,51)(15,38,35,21)(17,41,43,26)(18,46,49,31)(24,57,58,71)(27,45,34,53)(28,44,39,50)(30,48,68,54)(32,47,40,66)(42,61,69,64)(55,62,70,60)(59,65,63,72);(1,2,4,6)(3,8,7,5)(9,11,16,25)(10,17,22,43)(12,24,33,58)(13,34,36,27)(14,26,37,41)(15,39,35,28)(18,42,49,69)(19,50,51,44)(20,23,52,56)(21,53,38,45)(29,65,67,72)(30,62,68,60)(31,59,46,63)(32,64,40,61)(47,55,66,70)(48,57,54,71);(9,12,16,33)(10,18,22,49)(11,24,25,58)(13,31,36,46)(14,30,37,68)(15,40,35,32)(17,42,43,69)(19,29,51,67)(20,47,52,66)(21,54,38,48)(23,55,56,70)(26,62,41,60)(27,63,34,59)(28,64,39,61)(44,72,50,65)(45,57,53,71);(11,25)(17,43)(23,56)(24,58)(26,41)(27,34)(28,39)(42,69)(44,50)(45,53)(55,70)(57,71)(59,63)(60,62)(61,64)(65,72)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fhi,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=fhi,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, 396261);