#148025

order	512
nilpotency class	4
isoclinism	isoclinic class of #148023
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

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

normal subgroups only

This group has 813 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^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,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, 55221);