#114326

order	512
nilpotency class	3
isoclinism	isoclinic class of #114313
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

728 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,d^c=di,e^c=ei,f^c=fh,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, 21522);