#491143

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

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,c^b=ci,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, 398339);