#142402

order	512
nilpotency class	4
isoclinism	isoclinic class of #142402
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9921
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

366 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=gi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,g^a=gi,c^b=cf,d^b=dgi,f^b=fi,g^b=gi,h^b=hi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49598);