#135257

order	512
nilpotency class	4
isoclinism	isoclinic class of #12873
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1823
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

395 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 395 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=d,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,f^b=fi,e^c=eg,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237238, #72239680, #72239682
maximal normal subgroup of	#72237238, #72239680, #72239682

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42453);