#102782

order	512
nilpotency class	3
isoclinism	isoclinic class of #102749
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#392
commutator, Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

437 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 437 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,5,3)(4,6,8,7)(9,10,13,16)(11,17,19,24)(12,20,22,15)(14,21,23,18)(25,26)(27,31)(28,35)(29,37)(30,33)(32,34)(36,39)(38,40);(2,6)(3,7)(4,8)(9,11,13,19)(10,14,16,23)(12,21,22,18)(15,24,20,17)(25,27,30,29)(26,31,33,37)(28,34,36,38)(32,35,40,39);(1,3)(2,5)(4,6)(7,8)(12,22)(14,23)(15,20)(18,21)(25,28,30,36)(26,32,33,40)(27,34,29,38)(31,39,37,35)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=g,e^2=i,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fh,c^b=cf,d^b=dg,f^b=fghi,d^c=dh,e^c=ei,f^c=fghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9978);