#121217

order	512
nilpotency class	3
isoclinism	isoclinic class of #121215
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	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 337 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)(3,12)(4,17,21,27)(5,22)(6,10)(7,11)(8,14)(9,20,28,19)(13,29,31,23)(15,25)(16,24)(18,32,30,26)(33,34,38,44,39,42,37,43)(35,40,48,57,46,56,47,55)(36,49,51,58,52,59,53,41)(45,60,61,64,50,63,62,54);(1,3)(2,8,11,25)(4,18)(5,14)(6,15)(7,16)(9,26,28,32)(10,12,22,24)(13,19)(17,29,27,23)(20,31)(21,30)(33,35,37,48,39,46,38,47)(34,40,44,55,42,56,43,57)(36,50,51,61,52,45,53,62)(41,54,59,60,58,64,49,63);(1,4,5,19,7,21,6,20)(2,9,10,17,11,28,22,27)(3,13,14,30,16,31,15,18)(8,23,12,26,25,29,24,32)(33,36)(34,41,42,58)(35,45,46,50)(37,51)(38,53)(39,52)(40,54)(43,49,44,59)(47,61,48,62)(55,60)(56,64)(57,63)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=hi,e^2=h,f^2=hi,g^2=i,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,f^c=fh,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, 28413);