#118919

order	512
nilpotency class	3
isoclinism	isoclinic class of #118916
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
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

482 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26115);