#415113

order	512
nilpotency class	3
isoclinism	isoclinic class of #396100
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2414
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15183
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

841 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)(3,15,20,9)(4,21)(5,27)(6,14)(7,13)(8,12)(10,24)(11,29)(16,43,45,35)(17,46,49,56)(18,33,19,34)(22,41,50,31)(23,36)(25,37)(26,38)(28,39)(30,40)(32,47,42,48)(44,58,63,61)(51,54)(52,62,64,59)(53,57)(55,60);(1,3)(2,9)(4,22)(5,17)(6,19)(7,18)(8,20)(10,35)(11,32)(12,15)(13,34)(14,33)(16,24)(21,31)(23,52)(25,45)(26,50)(27,56)(28,48)(29,47)(30,49)(36,59)(37,43)(38,41)(39,42)(40,46)(44,53)(51,58)(54,63)(55,64)(57,61)(60,62);(1,4)(2,10,12,37)(3,16)(5,23)(6,24)(7,25)(8,26)(9,31,15,41)(11,36,39,60)(13,21,14,38)(17,44)(18,22)(19,50)(20,45)(27,57,40,51)(28,53)(29,54)(30,55)(32,58,42,61)(33,35,34,43)(46,62,56,59)(47,52)(48,64)(49,63);(1,5,7,29,8,30,6,28)(2,11,14,27,12,39,13,40)(3,17,18,47,20,49,19,48)(4,23,25,54,26,55,24,53)(9,32,33,56,15,42,34,46)(10,36,38,57,37,60,21,51)(16,44,22,52,45,63,50,64)(31,58,35,59,41,61,43,62)

Polycyclic presentation:

2^9|a^2=fh,c^2=e,d^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,g^c=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, 322309);