#148029

order	512
nilpotency class	4
isoclinism	isoclinic class of #148027
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11043
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

805 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 805 conjugacy classes of subgroups — too many to draw. Only its 132 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fi,h^a=hi,c^b=cf,f^b=fi,d^c=di,e^c=ei,g^c=ghi,g^e=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55225);