#118359

order	512
nilpotency class	3
isoclinism	isoclinic class of #118356
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

527 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:

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

Polycyclic presentation:

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

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, 25555);