#118498

order	512
nilpotency class	3
isoclinism	isoclinic class of #118493
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

544 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

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, 25694);