#491131

order	512
nilpotency class	4
isoclinism	isoclinic class of #491091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂³	#466
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

387 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 387 conjugacy classes of subgroups — too many to draw. Only its 128 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 — 4 generators on 80 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398327);