#121099

order	512
nilpotency class	3
isoclinism	isoclinic class of #121099
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1443
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

348 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,e^b=ei,g^b=ghi,e^c=eh,f^c=fhi,g^c=gh

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, 28295);