#135892

order	512
nilpotency class	4
isoclinism	isoclinic class of #2190
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#447
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8843
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

292 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 102 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=gi,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dg,e^b=egi,g^b=gi,d^c=dgi,e^c=eg,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43088);