#114234

order	512
nilpotency class	3
isoclinism	isoclinic class of #114229
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

741 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:

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

Polycyclic presentation:

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

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, 21430);