#144377

order	512
nilpotency class	4
isoclinism	isoclinic class of #144375
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂²×#41	C₂³
prime radical	#3487	C₂²
LCS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

835 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 835 conjugacy classes of subgroups — too many to draw. Only its 56 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)(4,6)(7,8)(9,10)(11,22)(12,28)(13,34)(14,38)(15,20)(16,21)(17,24)(18,31)(19,37)(23,54)(25,59)(26,53)(27,47)(29,62)(30,64)(32,46)(33,51)(35,43)(36,49)(39,50)(40,63)(41,60)(42,58)(44,69)(45,66)(48,68)(52,56)(55,70)(57,72)(61,67)(65,71);(5,8)(9,11,16,27)(10,17,21,43)(12,29,33,56)(13,35,37,24)(14,26,15,25)(18,44,46,70)(19,47,34,22)(20,41,38,42)(23,31,57,32)(28,40,51,71)(30,65,67,63)(36,58,39,60)(45,54,48,72)(49,59,50,53)(52,64,62,61)(55,66,69,68);(1,3)(2,5)(4,7)(6,8)(9,12,16,33)(10,18,21,46)(11,23,27,57)(13,30,37,67)(14,32,15,31)(17,40,43,71)(19,48,34,45)(20,51,38,28)(22,52,47,62)(24,55,35,69)(25,29,26,56)(36,68,39,66)(41,44,42,70)(49,64,50,61)(53,72,59,54)(58,65,60,63)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fi,d^c=dg,e^c=ei,f^c=fi,e^d=ei

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51573);