#214655

order	512
nilpotency class	3
isoclinism	isoclinic class of #214040
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

777 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=hi,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=di,f^b=fh,g^b=gh,f^c=fh,g^c=ghi,g^d=gh

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, 121851);