#121267

order	512
nilpotency class	3
isoclinism	isoclinic class of #121267
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

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

normal subgroups only

This group has 354 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) = 64

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

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

Polycyclic presentation:

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

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, 28463);