#118251

order	512
nilpotency class	3
isoclinism	isoclinic class of #118246
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

531 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,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,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25447);