#118542

order	512
nilpotency class	3
isoclinism	isoclinic class of #118537
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

545 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=hi,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,f^b=fhi,d^c=di,e^c=ehi,f^c=fh,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, 25738);