#129280

order	512
nilpotency class	3
isoclinism	isoclinic class of #8534
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1779
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

632 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 118 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,21,26,22,27,36,46,23,28,37,47,38,48,59,69)(19,24,31,41,32,42,55,65,30,40,53,63,54,64,74,78)(20,25,34,44,35,45,58,68,33,43,56,66,57,67,75,79)(29,49,51,70,52,71,73,80,50,39,60,61,72,62,76,77);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,24)(20,29)(21,31)(22,32)(23,30)(25,39)(26,41)(27,42)(28,40)(33,50)(34,51)(35,52)(36,55)(37,53)(38,54)(43,49)(44,61)(45,62)(46,65)(47,63)(48,64)(56,60)(57,72)(58,73)(59,74)(66,70)(67,71)(68,77)(69,78)(75,76)(79,80);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,25)(19,29)(21,34)(22,35)(23,33)(24,39)(26,44)(27,45)(28,43)(30,50)(31,51)(32,52)(36,58)(37,56)(38,57)(40,49)(41,61)(42,62)(46,68)(47,66)(48,67)(53,60)(54,72)(55,73)(59,75)(63,70)(64,71)(65,77)(69,79)(74,76)(78,80)

Polycyclic presentation:

2^9|a^2=f,c^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^b=ei,f^b=fg,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, 36476);