#110186

order	512
nilpotency class	3
isoclinism	isoclinic class of #110180
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

533 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 533 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,f^2=hi,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17382);