#146257

order	512
nilpotency class	4
isoclinism	isoclinic class of #146219
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2092
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

291 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 291 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

2^9|a^2=dh,b^2=de,d^2=gi,e^2=gi,f^2=hi,g^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=dg,e^b=egi,f^b=fh,g^b=gi,d^c=dg,e^c=eg,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, 53453);