#140588

order	512
nilpotency class	3
isoclinism	isoclinic class of #10370
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1955
prime radical	#532	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

335 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 335 conjugacy classes of subgroups — too many to draw. Only its 94 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)(9,10,13,19,15,20,31,44,14,18,30,33,32,42,60,62)(11,16,23,36,24,37,51,66,22,35,29,43,50,65,59,71)(12,17,27,39,28,41,57,63,26,40,53,61,55,69,46,72)(21,45,48,70,49,38,56,67,47,34,58,64,25,52,54,68);(1,2,3,5,4,6,7,8)(9,11,13,29,15,24,31,59,14,22,30,23,32,50,60,51)(10,16,33,36,20,37,62,66,18,35,19,43,42,65,44,71)(12,25,53,54,28,21,46,48,26,49,27,56,55,47,57,58)(17,38,39,68,41,34,63,70,40,52,61,67,69,45,72,64);(2,6)(5,8)(9,12)(10,17)(11,21,22,47)(13,27)(14,26)(15,28)(16,34,35,45)(18,40)(19,39)(20,41)(23,48,29,58)(24,49,50,25)(30,53)(31,57)(32,55)(33,61)(36,64,43,70)(37,52,65,38)(42,69)(44,63)(46,60)(51,56,59,54)(62,72)(66,68,71,67)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=e,d^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=cgh,d^b=di,e^b=ei,f^b=fi,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, 47784);