#145002

order	512
nilpotency class	4
isoclinism	isoclinic class of #145002
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2033
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#28497	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

403 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 403 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 128

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

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

Polycyclic presentation:

2^9|c^2=d,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fh,g^a=gi,c^b=cf,d^b=di,e^b=eh,f^b=fh,e^c=egi,f^c=fhi,g^c=gi

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, 52198);