#135752

order	512
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1843
prime radical	C₂³	#424
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

208 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 98 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)(3,9)(4,12)(5,7)(6,10)(8,13)(11,15)(14,16)(17,18,21,26,22,27,37,48,23,28,38,49,39,50,63,74)(19,24,31,42,32,43,56,68,30,41,54,66,55,67,76,79)(20,25,35,46,36,47,62,73,34,45,60,71,61,72,78,80)(29,51,53,75,33,57,59,77,52,40,64,65,58,44,69,70);(1,3,4,10,5,11,13,16)(2,6,8,9,7,14,12,15)(17,19)(18,24)(20,33,34,58)(21,31)(22,32)(23,30)(25,44,45,57)(26,42)(27,43)(28,41)(29,36,52,61)(35,59,60,69)(37,56)(38,54)(39,55)(40,47,51,72)(46,70,71,77)(48,68)(49,66)(50,67)(53,62,64,78)(63,76)(65,73,75,80)(74,79);(2,7)(6,14)(8,12)(9,15)(17,20)(18,25)(19,29,30,52)(21,35)(22,36)(23,34)(24,40,41,51)(26,46)(27,47)(28,45)(31,53,54,64)(32,33,55,58)(37,62)(38,60)(39,61)(42,65,66,75)(43,44,67,57)(48,73)(49,71)(50,72)(56,59,76,69)(63,78)(68,70,79,77)(74,80)

Polycyclic presentation:

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

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, 42948);