#412347

order	512
nilpotency class	3
isoclinism	isoclinic class of #401790
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2485
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15129
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1221 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:

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

Polycyclic presentation:

2^9|a^2=ef,d^2=h,e^2=i,g^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,e^b=ei,f^b=fi,g^b=gi,e^c=ei,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, 319543);