#121114

order	512
nilpotency class	3
isoclinism	isoclinic class of #121113
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#117
LCS	G ▷ C₂³×C₄ ▷ 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:

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

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, 28310);