#121070

order	512
nilpotency class	3
isoclinism	isoclinic class of #121070
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	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

336 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 104 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28266);