#97246

order	512
nilpotency class	3
isoclinism	isoclinic class of #97214
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

474 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,22,28,23,27,21,26)(19,29,32,24,33,40,31,39)(20,34,35,42,36,25,37,41)(30,44,45,38,46,43,47,48)(49,50)(51,59,62,72)(52,63,65,56)(53,57)(54,58)(55,61,70,66)(60,75)(64,73,67,71)(68,74)(69,76)(77,80)(78,79);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,23,33)(18,24,27,39)(20,30,36,46)(21,32,22,31)(25,38,34,48)(26,40,28,29)(35,47,37,45)(41,44,42,43)(49,51,54,62)(50,55,58,70)(52,60,65,77)(53,66,68,61)(56,69,63,79)(57,72,74,59)(64,78,67,76)(71,75,73,80);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,41)(27,34)(28,42)(29,43)(31,45)(32,47)(33,46)(39,48)(40,44)(49,52)(50,56)(51,60)(53,67)(54,65)(55,69)(57,73)(58,63)(59,75)(61,78)(62,77)(64,68)(66,76)(70,79)(71,74)(72,80)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=g,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=ei,f^b=fgh,d^c=di,f^c=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4442);