#148010

order	512
nilpotency class	4
isoclinism	isoclinic class of #148009
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	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂³×D₄	C₂³
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 896 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,g^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,f^b=fi,d^c=di,g^c=gh,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, 55206);