#135757

order	512
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1843
prime radical	C₂³	#424
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

218 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 102 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=e,d^2=g,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cgh,e^b=ei,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42953);