#507092

order	512
nilpotency class	4
isoclinism	isoclinic class of #501994
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3341
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#28828
prime radical	C₂³	#578
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

317 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=eg,c^2=ef,d^2=e,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414288);