#143167

order	512
nilpotency class	4
isoclinism	isoclinic class of #143166
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10018
commutator, Frattini, cosocle	C₄×M₁₆	C₂³
prime radical	#33688	C₂
LCS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

370 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 50 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50363);