#116773

order	512
nilpotency class	3
isoclinism	isoclinic class of #116768
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
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

522 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=hi,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,e^b=eh,f^b=fi,e^c=ehi,f^c=fi,g^c=gh

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, 23969);