#120898

order	512
nilpotency class	3
isoclinism	isoclinic class of #120896
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	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

354 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 112 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)(3,11)(4,8,17,24)(5,18)(6,10)(7,13)(9,19)(12,25,15,29)(14,26)(16,31,32,23)(20,30,22,27)(21,28)(33,34,37,42,38,41,52,58)(35,43,46,60,45,39,53,55)(36,47,50,59,49,40,51,57)(44,54,61,63,48,56,62,64);(1,3)(2,7,10,21)(4,15,17,12)(5,13)(6,14)(8,22)(9,11,18,26)(16,30,32,27)(19,28)(20,24)(23,29)(25,31)(33,35)(34,39,41,43)(36,48)(37,46)(38,45)(40,56,47,54)(42,55,58,60)(44,49)(50,62)(51,61)(52,53)(57,64,59,63);(1,4,5,16,6,17,19,32)(2,8,9,23,10,24,18,31)(3,12,13,27,14,15,28,30)(7,20,11,25,21,22,26,29)(33,36)(34,40)(35,44,45,48)(37,51)(38,49)(39,54,43,56)(41,47)(42,59)(46,62,53,61)(50,52)(55,64,60,63)(57,58)

Polycyclic presentation:

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

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, 28094);