#118347

order	512
nilpotency class	3
isoclinism	isoclinic class of #118344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

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

535 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:

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

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, 25543);