#102689

order	512
nilpotency class	3
isoclinism	isoclinic class of #102685
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₄³	C₂³
prime radical	#30268	C₂
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

497 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9885);