#388017

order	512
nilpotency class	3
isoclinism	isoclinic class of #32700
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2760
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14508
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

757 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=eg,d^2=eg,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 295213);