#102666

order	512
nilpotency class	3
isoclinism	isoclinic class of #102649
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

499 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 56 points:

(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,17)(16,21)(18,22)(19,24)(20,23)(25,26,31,36)(27,37,40,48)(28,33,44,51)(29,35,30,34)(32,41,47,39)(38,53,54,49)(42,56,55,46)(43,52,45,50);(5,8)(9,11,13,19)(10,14,17,24)(12,20,21,18)(15,23,16,22)(25,27)(26,32)(28,42)(29,41)(30,39)(31,40)(33,49)(34,48)(35,37)(36,47)(38,43)(44,55)(45,54)(46,50)(51,53)(52,56);(1,3)(2,5)(4,7)(6,8)(10,15)(12,21)(14,22)(16,17)(18,20)(23,24)(25,28,31,44)(26,33,36,51)(27,38,40,54)(29,43,30,45)(32,46,47,56)(34,50,35,52)(37,49,48,53)(39,42,41,55)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=g,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fghi,c^b=cf,d^b=dg,e^b=egi,f^b=fg,d^c=dh,e^c=ei,f^c=fg

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9862);