#145026

order	512
nilpotency class	4
isoclinism	isoclinic class of #145026
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#28495	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

384 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|c^2=dg,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fhi,g^a=gi,c^b=cf,d^b=di,e^b=eh,f^b=fh,d^c=di,e^c=eg,f^c=fhi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52222);