#145966

order	512
nilpotency class	4
isoclinism	isoclinic class of #144056
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1697
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10599
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

244 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=deh,b^2=dg,d^2=gi,e^2=h,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,g^a=gi,h^a=hi,c^b=cf,d^c=dgi,e^c=eh,f^c=fg,g^c=gi

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, 53162);