#134492

order	512
nilpotency class	4
isoclinism	isoclinic class of #124660
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1510
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8575
prime radical	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

386 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

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

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, 41688);