#152309

order	512
nilpotency class	3
isoclinism	isoclinic class of #1828
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#381
commutator	C₂³	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#12844
prime radical	C₂⁴	C₂×C₁₆
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

289 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 289 conjugacy classes of subgroups — too many to draw. Only its 112 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=d,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^b=fi,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59505);