#434711

order	512
nilpotency class	3
isoclinism	isoclinic class of #420517
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15506
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1041 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 341907);