#121146

order	512
nilpotency class	3
isoclinism	isoclinic class of #121143
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	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

340 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 104 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)(3,12,15,8)(4,16)(5,20)(6,11)(7,10)(9,18)(13,30,32,26)(14,24,21,25)(17,23,31,29)(19,28)(22,27)(33,34,37,42,38,41,52,57)(35,43,46,60,45,39,53,55)(36,47,50,58,49,40,51,56)(44,59,61,64,48,54,62,63);(1,3,5,14,7,15,6,21)(2,8,20,25,10,12,11,24)(4,17,18,13,19,31,22,32)(9,26,28,23,27,30,16,29)(33,35)(34,39,41,43)(36,48)(37,46)(38,45)(40,54,47,59)(42,55,57,60)(44,49)(50,62)(51,61)(52,53)(56,63,58,64);(1,4)(2,9,10,27)(3,13)(5,18)(6,22)(7,19)(8,23,12,29)(11,16,20,28)(14,31)(15,32)(17,21)(24,26,25,30)(33,36)(34,40)(35,44,45,48)(37,51)(38,49)(39,54,43,59)(41,47)(42,58)(46,62,53,61)(50,52)(55,64,60,63)(56,57)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28342);