#129672

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

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	#1779
prime radical	#3531	C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

519 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 519 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 256

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

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

Polycyclic presentation:

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

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, 36868);