#148027

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

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

normal subgroups only

This group has 821 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)(3,12)(4,16)(5,10,19,8)(6,11)(7,14)(9,17)(13,26)(15,25,28,31)(18,23,30,24)(20,29,21,32)(22,27);(1,3,4,14,6,13,17,27)(2,7,9,12,11,22,16,26)(5,15,18,29,19,28,30,32)(8,20,23,31,10,21,24,25);(2,8)(3,13)(4,17)(5,19)(7,20)(9,24)(10,11)(12,25)(16,23)(21,22)(26,31)(29,32)

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,e^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, 55223);