#142377

order	512
nilpotency class	4
isoclinism	isoclinic class of #142246
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

386 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 70 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 — 3 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=ei,g^b=gi,d^c=dh,e^c=ei,f^c=fi

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, 49573);