#146219

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:

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

Polycyclic presentation:

2^9|a^2=d,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, 53415);