#139375

order	512
nilpotency class	4
isoclinism	isoclinic class of #139356
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

307 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 80 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 — 3 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=ghi,e^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=ei,f^c=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46571);