#488634

order	512
nilpotency class	4
isoclinism	isoclinic class of #488553
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2922 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

704 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 130 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 — 4 generators on 80 points:

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

Polycyclic presentation:

2^9|c^2=i,d^2=h,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eg,f^a=fi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395830);