#442357

order	512
nilpotency class	3
isoclinism	isoclinic class of #421157
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2610
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15709
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

859 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)(3,14,18,34)(4,19,22,10)(5,24)(6,12)(7,30)(8,13)(9,17,33,29)(11,27)(15,40)(16,32,44,58)(20,48,51,54)(21,47,23,36)(25,37,56,59)(26,38)(28,55)(31,42)(35,52,46,50)(39,43)(41,62)(45,53)(49,61)(57,63)(60,64);(1,3,6,29,8,18,7,17)(2,9,12,34,13,33,30,14)(4,15,23,42,22,43,21,53)(5,25,26,16,28,56,27,44)(10,31,47,39,19,45,36,40)(11,37,24,32,38,59,55,58)(20,49,52,63,51,64,50,41)(35,60,48,62,46,61,54,57);(1,4,6,21,8,22,7,23)(2,10,12,36,13,19,30,47)(3,15,17,42,18,43,29,53)(5,20,26,50,28,51,27,52)(9,31,14,39,33,45,34,40)(11,35,24,54,38,46,55,48)(16,41,25,49,44,63,56,64)(32,57,37,60,58,62,59,61);(1,5,6,26,8,28,7,27)(2,11,30,55,13,38,12,24)(3,16,17,25,18,44,29,56)(4,20,21,50,22,51,23,52)(9,32,34,59,33,58,14,37)(10,35,47,48,19,46,36,54)(15,41,42,49,43,63,53,64)(31,57,40,61,45,62,39,60)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,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, 349553);