#146232

order	512
nilpotency class	4
isoclinism	isoclinic class of #146231
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2094
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#492
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

287 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=dg,b^2=de,c^2=i,d^2=g,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,d^b=dg,e^b=egi,f^b=fh,g^b=gi,h^b=hi,d^c=dgi,e^c=egi,f^c=fhi,g^c=gi,h^c=hi

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, 53428);