#102679

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

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

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

normal subgroups only

This group has 409 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,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, 9875);