#142803

order	512
nilpotency class	4
isoclinism	isoclinic class of #142801
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×#117	C₂³
prime radical	#3449	C₂²
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

636 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49999);