#118387

order	512
nilpotency class	3
isoclinism	isoclinic class of #118383
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	#117
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

508 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)(3,13)(4,17)(5,11)(6,10)(7,12)(8,15)(9,18)(14,22)(16,26)(19,27)(20,23)(21,25)(24,31)(28,30)(29,32)(33,34,37,42,38,41,51,56)(35,39,45,54,44,53,60,63)(36,46,49,57,48,40,50,55)(43,58,59,64,47,52,61,62);(1,3,5,15,7,16,6,20)(2,8,10,13,12,23,11,26)(4,14,18,29,19,31,21,30)(9,24,17,32,25,22,27,28)(33,35)(34,39)(36,47,48,43)(37,45)(38,44)(40,52,46,58)(41,53)(42,54)(49,61,50,59)(51,60)(55,62,57,64)(56,63);(1,4,7,19)(2,9,12,25)(3,14,16,31)(5,18,6,21)(8,22,23,24)(10,17,11,27)(13,28,26,32)(15,29,20,30)(33,36)(34,40)(35,43)(37,50)(38,48)(39,52)(41,46)(42,57)(44,47)(45,61)(49,51)(53,58)(54,64)(55,56)(59,60)(62,63)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,c^b=cf,e^b=eh,f^b=fi,d^c=di,e^c=eh,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, 25583);