#119043

order	512
nilpotency class	3
isoclinism	isoclinic class of #119038
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

489 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)(3,13,16,24,17,8,15,25)(4,18,20,27,22,28,21,9)(14,30,31,23,32,26,19,29)(33,34)(35,44)(36,49)(37,54)(38,42)(39,43)(40,46)(41,51)(45,55)(47,57)(48,56)(50,58)(52,59)(53,62)(60,63)(61,64);(1,3,7,17)(2,8,10,13)(4,19,22,31)(5,16,6,15)(9,26,27,30)(11,25,12,24)(14,20,32,21)(18,23,28,29)(33,35,39,48)(34,40,43,57)(36,50,53,60)(37,47,38,46)(41,58,59,63)(42,56,54,44)(45,51,61,52)(49,55,62,64);(1,4)(2,9,10,27)(3,14)(5,20)(6,21)(7,22)(8,23,13,29)(11,18,12,28)(15,31)(16,19)(17,32)(24,26,25,30)(33,36,37,51,39,53,38,52)(34,41,42,49,43,59,54,62)(35,45,46,60,48,61,47,50)(40,55,44,58,57,64,56,63)

Polycyclic presentation:

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

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, 26239);