#140122

order	512
nilpotency class	4
isoclinism	isoclinic class of #2238
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#481
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9542
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

245 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 245 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225462, #72225467, #72225469
maximal normal subgroup of	#72225462, #72225467, #72225469

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47318);