#507093

order	512
nilpotency class	4
isoclinism	isoclinic class of #501995
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

315 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 315 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

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,d^b=di,g^c=gh,h^c=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, 414289);