#121095

order	512
nilpotency class	3
isoclinism	isoclinic class of #121095
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

362 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 64

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

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

Polycyclic presentation:

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

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, 28291);