#435484

order	512
nilpotency class	3
isoclinism	isoclinic class of #419464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2557
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15527
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

911 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)(4,20,25,41)(5,26)(6,13)(7,14)(8,12)(10,23,37,24)(11,27)(16,35)(17,46,49,55)(18,34,30,33)(21,42)(22,36,53,61)(28,39)(29,40)(31,44)(32,48,38,47)(43,57)(45,58)(50,54,56,52)(51,59)(60,63)(62,64);(1,3,6,18,8,19,7,30)(2,9,13,33,12,15,14,34)(4,21,23,16,25,44,24,45)(5,17,27,47,29,49,28,48)(10,35,20,31,37,58,41,42)(11,38,40,46,39,32,26,55)(22,51,52,43,53,63,54,64)(36,60,50,62,61,59,56,57);(1,4,6,23,8,25,7,24)(2,10,14,41,12,37,13,20)(3,16,18,44,19,45,30,21)(5,22,27,52,29,53,28,54)(9,31,34,35,15,42,33,58)(11,36,26,56,39,61,40,50)(17,43,47,63,49,64,48,51)(32,57,46,59,38,62,55,60);(1,5)(2,11)(3,17)(4,22)(6,27)(7,28)(8,29)(9,32)(10,36)(12,39)(13,40)(14,26)(15,38)(16,43)(18,47)(19,49)(20,50)(21,51)(23,52)(24,54)(25,53)(30,48)(31,57)(33,55)(34,46)(35,59)(37,61)(41,56)(42,62)(44,63)(45,64)(58,60)

Polycyclic presentation:

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

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, 342680);