#121104

order	512
nilpotency class	3
isoclinism	isoclinic class of #121103
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:

(1,2,6,10,7,12,5,11)(3,13,16,24,17,28,15,8)(4,18,20,9,22,26,21,27)(14,30,19,29,32,23,31,25)(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)(2,8,12,24)(4,19)(5,15)(6,16)(7,17)(9,25,27,29)(10,13,11,28)(14,21)(18,23,26,30)(20,32)(22,31)(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);(1,4,5,20,7,22,6,21)(2,9,10,18,12,27,11,26)(3,14,15,19,17,32,16,31)(8,23,13,29,24,30,28,25)(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=hi,e^2=h,f^2=h,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=fh,g^b=gi,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, 28300);