#146067

order	512
nilpotency class	4
isoclinism	isoclinic class of #146024
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2076
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#469
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

200 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 70 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)(3,6)(4,7)(5,8)(9,16)(10,14)(11,13)(12,15)(17,18,24,46,43,59,42,58,23,29,22,28,21,27,48,72)(19,30,44,77,32,53,69,51,35,52,33,63,34,25,49,60)(20,36,70,55,41,76,45,26,40,73,39,57,38,61,47,56)(31,64,75,54,65,71,68,62,37,74,66,50,67,80,78,79);(1,3,4,10)(2,6,7,14)(5,9,12,11)(8,16,15,13)(17,19)(18,25)(20,37)(21,34)(22,44)(23,35)(24,33)(26,54)(27,52)(28,60)(29,53)(30,59)(31,40)(32,43)(36,71)(38,65)(39,66)(41,67)(42,49)(45,68)(46,51)(47,78)(48,69)(50,56)(55,79)(57,62)(58,63)(61,64)(70,75)(72,77)(73,80)(74,76);(2,7)(3,9)(5,12)(6,13)(10,11)(14,16)(17,20,23,40)(18,26,29,56)(19,31)(21,41,43,38)(22,45,24,47)(25,50)(27,57,59,55)(28,61,46,76)(30,62)(32,67)(33,68)(34,65)(35,37)(36,72,73,58)(39,48,70,42)(44,78)(49,75)(51,64)(52,79)(53,54)(60,74)(63,80)(66,69)(71,77)

Polycyclic presentation:

2^9|a^2=ehi,b^2=h,c^2=dg,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=di,e^b=ei,f^b=fh,d^c=di,e^c=eg,f^c=fhi,g^c=gi

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, 53263);