#129285

order	512
nilpotency class	3
isoclinism	isoclinic class of #8453
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	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1779
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 628 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)(17,18,22,28,23,29,43,56,24,30,44,57,21,27,41,54)(19,31,35,59,33,47,65,55,34,52,42,70,39,25,45,48)(20,26,37,50,36,49,66,75,38,51,68,77,40,53,69,78)(32,60,63,80,64,46,72,74,62,58,67,79,61,73,71,76);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,25)(20,32)(21,39)(22,42)(23,33)(24,34)(26,46)(27,52)(28,55)(29,31)(30,47)(35,44)(36,64)(37,67)(38,62)(40,61)(41,65)(43,45)(48,57)(49,58)(50,76)(51,73)(53,60)(54,59)(56,70)(63,68)(66,71)(69,72)(74,77)(75,80)(78,79);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,26)(19,32)(21,40)(22,37)(23,36)(24,38)(25,46)(27,53)(28,50)(29,49)(30,51)(31,58)(33,64)(34,62)(35,63)(39,61)(41,69)(42,67)(43,66)(44,68)(45,71)(47,73)(48,74)(52,60)(54,78)(55,76)(56,75)(57,77)(59,79)(65,72)(70,80)

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=di,e^b=ei,f^b=fgi,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, 36481);