#134615

order	512
nilpotency class	4
isoclinism	isoclinic class of #134475
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1804
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

360 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 100 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:

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

Polycyclic presentation:

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

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, 41811);