#145148

order	512
nilpotency class	4
isoclinism	isoclinic class of #145147
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	#532	D₄
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

331 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=i,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=fi,g^a=gi,c^b=cf,d^b=di,f^b=fh,d^c=di,e^c=eg,f^c=fhi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52344);