#116522

order	512
nilpotency class	3
isoclinism	isoclinic class of #116517
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

511 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^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,e^b=ehi,f^b=fh,f^c=fh,g^c=gi

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, 23718);