#110184

order	512
nilpotency class	3
isoclinism	isoclinic class of #110180
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₄²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

547 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 547 conjugacy classes of subgroups — too many to draw. Only its 144 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72235180
maximal subgroup of	#72235160, #72235180
maximal normal subgroup of	#72235160, #72235180

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17380);