#121301

order	512
nilpotency class	3
isoclinism	isoclinic class of #121298
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	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 335 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) = 128

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

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

Polycyclic presentation:

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

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, 28497);