#479852

order	512
nilpotency class	3
isoclinism	isoclinic class of #478580
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2789
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17280
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 535 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387048);