#214653

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

757 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)(3,11,14,7)(4,15,17,27)(5,10)(6,9)(8,16,24,19)(12,28)(13,26,18,22)(20,25)(21,29)(23,31)(30,32)(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)(2,7)(4,12)(5,18)(6,14)(8,23)(9,11)(10,26)(13,20)(15,32)(16,31)(17,30)(19,29)(21,24)(22,25)(27,28)(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);(1,4,6,17)(2,8,9,24)(3,12,14,30)(5,19,20,16)(7,21,11,23)(10,27,25,15)(13,31,18,29)(22,28,26,32)(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);(1,3)(2,7)(4,12)(5,13)(6,14)(8,21)(9,11)(10,22)(15,32)(16,29)(17,30)(18,20)(19,31)(23,24)(25,26)(27,28)(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=i,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, 121849);