#113885

order	512
nilpotency class	3
isoclinism	isoclinic class of #113877
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	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

734 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=i,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,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=dhi,e^c=ehi,f^c=fh,g^c=gi

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, 21081);