#136800

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1869
prime radical	#532	C₂×C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

312 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 80
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 80 points:

(1,2)(3,11,14,15)(4,8,7,13)(5,9)(6,10)(12,16)(17,18,22,28,23,29,43,56,24,30,44,57,21,27,41,54)(19,25,33,47,32,46,63,74,34,48,65,76,39,52,69,78)(20,35,38,67,37,50,68,55,36,53,42,70,40,26,49,51)(31,58,61,79,62,45,71,73,59,66,64,80,60,72,77,75);(1,3,12,13,6,14,5,8)(2,7,16,11,10,4,9,15)(17,19,22,33,23,32,43,63,24,34,44,65,21,39,41,69)(18,25,28,47,29,46,56,74,30,48,57,76,27,52,54,78)(20,31,42,64,37,62,49,77,36,59,38,61,40,60,68,71)(26,45,55,75,35,66,70,79,50,72,51,73,53,58,67,80);(1,4,6,7)(2,8,10,13)(3,9,14,16)(5,11,12,15)(17,20)(18,26)(19,31)(21,40)(22,42)(23,37)(24,36)(25,45)(27,53)(28,55)(29,35)(30,50)(32,62)(33,64)(34,59)(38,44)(39,60)(41,68)(43,49)(46,66)(47,75)(48,72)(51,57)(52,58)(54,67)(56,70)(61,65)(63,77)(69,71)(73,76)(74,79)(78,80)

Polycyclic presentation:

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

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, 43996);