#118926

order	512
nilpotency class	3
isoclinism	isoclinic class of #118926
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

496 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)(3,11,13,7)(4,15)(5,19)(6,9)(8,17)(10,22)(12,29,30,23)(14,24,20,27)(16,25,31,32)(18,26)(21,28)(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)(2,7)(4,16)(5,20)(6,13)(8,25)(9,11)(10,27)(12,18)(14,22)(15,32)(17,31)(19,24)(21,30)(23,26)(28,29)(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)(2,8,9,15)(3,12)(5,21)(6,17)(7,23,11,29)(10,28,19,26)(13,30)(14,31)(16,20)(18,22)(24,32,27,25)(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,c^2=d,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,e^c=eh,f^c=fhi,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, 26122);