#471110

order	512
nilpotency class	3
isoclinism	isoclinic class of #460970
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2759
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16755
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

515 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)(3,15,19,9)(4,21)(5,26)(6,14)(7,13)(8,12)(10,24)(11,28)(16,43,22,52)(17,46,48,56)(18,33,20,34)(23,38)(25,41)(27,39)(29,40)(30,37)(31,45,35,50)(32,49,42,47)(36,55)(44,58,53,59)(51,54)(57,60)(61,63,64,62);(1,3)(2,9)(4,22,25,16)(5,17)(6,18)(7,20)(8,19)(10,35,37,31)(11,32)(12,15)(13,33)(14,34)(21,43,41,52)(23,53,55,44)(24,50,30,45)(26,56)(27,47)(28,49)(29,48)(36,59,38,58)(39,42)(40,46)(51,61,60,64)(54,63,57,62);(1,4)(2,10,12,37)(3,16)(5,23)(6,24)(7,30)(8,25)(9,31,15,35)(11,38,39,36)(13,21,14,41)(17,44)(18,45)(19,22)(20,50)(26,51,40,60)(27,54)(28,57)(29,55)(32,59,42,58)(33,52,34,43)(46,61,56,64)(47,62)(48,53)(49,63);(1,5,6,27,8,29,7,28)(2,11,13,40,12,39,14,26)(3,17,18,47,19,48,20,49)(4,23,24,54,25,55,30,57)(9,32,33,46,15,42,34,56)(10,36,21,51,37,38,41,60)(16,44,45,62,22,53,50,63)(31,58,52,64,35,59,43,61)

Polycyclic presentation:

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

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, 378306);