#152239

order	512
nilpotency class	5
isoclinism	isoclinic class of #149794
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2126
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#532	C₂³
socle	C₂	#12797
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

372 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,e^c=ehi,f^c=fgh,g^c=gi,f^e=fhi,g^f=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, 59435);