#120904

order	512
nilpotency class	3
isoclinism	isoclinic class of #120904
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
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

378 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 114 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)(3,15,19,36)(4,8)(5,12)(6,13)(7,14)(9,18,35,20)(10,23)(11,27)(16,44,47,63)(17,49,52,33)(21,31)(22,30)(24,38)(25,40)(26,41)(28,51,43,53)(29,54)(32,60,61,50)(34,46,58,48)(37,56)(39,57)(42,64,62,45)(55,59);(2,8,11,29)(3,16,18,46)(4,21)(5,24)(9,32,15,42)(10,14,31,13)(12,39,38,30)(17,50,51,64)(19,47,20,48)(22,56)(23,54)(25,26)(28,58,49,63)(33,44,43,34)(35,61,36,62)(37,40,59,41)(45,52,60,53)(55,57);(1,3,4,17,6,18,23,51)(2,9,10,33,13,36,8,28)(5,16,22,45,25,46,56,60)(7,20,21,53,27,19,54,52)(11,35,31,49,14,15,29,43)(12,34,37,62,40,63,30,32)(24,47,57,64,26,48,55,50)(38,58,59,42,41,44,39,61)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,f^b=fhi,e^c=eh,g^c=ghi

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, 28100);