#149608

order	512
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#99
commutator	C₂³	C₂²×C₁₆
Frattini, cosocle	C₂×M₃₂	C₂³
socle	C₂	C₂×#1322
prime radical	C₂⁴	C₂×C₁₆
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

251 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 251 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56804);