#141187

order	512
nilpotency class	4
isoclinism	isoclinic class of #140953
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1964
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

420 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=e,d^2=ghi,e^2=i,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fhi,g^b=gi,h^b=hi,d^c=di,f^d=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, 48383);