#437572

order	512
nilpotency class	3
isoclinism	isoclinic class of #419463
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2557
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15603
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

684 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)(4,20,24,41)(5,11)(6,13)(7,14)(8,12)(10,23,35,29)(16,33)(17,31,47,36)(18,40,28,32)(21,42)(22,48,51,62)(25,38)(26,39)(27,37)(30,44)(34,50,59,53)(43,58)(45,55)(46,56,52,61)(49,54)(57,64)(60,63);(1,3,6,18,8,19,7,28)(2,9,13,32,12,15,14,40)(4,21,23,16,24,44,29,45)(5,17,25,46,27,47,26,52)(10,33,20,30,35,55,41,42)(11,36,38,61,37,31,39,56)(22,49,50,43,51,63,53,64)(34,58,48,60,59,57,62,54);(1,4,6,23,8,24,7,29)(2,10,14,41,12,35,13,20)(3,16,18,44,19,45,28,21)(5,22,25,50,27,51,26,53)(9,30,40,33,15,42,32,55)(11,34,39,62,37,59,38,48)(17,43,46,63,47,64,52,49)(31,54,61,57,36,60,56,58);(1,5,6,25,8,27,7,26)(2,11,13,38,12,37,14,39)(3,17,18,46,19,47,28,52)(4,22,23,50,24,51,29,53)(9,31,32,56,15,36,40,61)(10,34,20,48,35,59,41,62)(16,43,44,63,45,64,21,49)(30,54,55,58,42,60,33,57)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=ef,d^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,g^b=gi,h^b=hi,g^c=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, 344768);