#143738

order	512
nilpotency class	4
isoclinism	isoclinic class of #143705
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#449
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

397 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 256

A minimal faithful permutation representation — 3 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=g,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dg,f^b=fi,d^c=di,e^c=eh,f^c=fi,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, 50934);