#140821

order	512
nilpotency class	3
isoclinism	isoclinic class of #113366
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1955
prime radical	C₂³	#435
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

250 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 256

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=e,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=ch,d^b=dg,e^b=ei,f^b=fg,d^c=di

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, 48017);