#142398

order	512
nilpotency class	4
isoclinism	isoclinic class of #142398
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9920
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	#29751	C₂
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 467 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)(3,12)(4,17)(5,6)(9,25)(10,18)(11,24)(13,22)(14,31)(15,16)(19,20)(21,23)(26,27)(28,30);(1,2,6,21,7,11,5,10)(3,13,12,28,16,27,15,9)(4,18,17,24,20,23,19,8)(14,22,29,25,31,26,32,30);(1,3)(2,9)(4,14)(5,15)(6,12)(7,16)(8,22)(10,27)(11,28)(13,21)(17,29)(18,25)(19,32)(20,31)(23,30)(24,26)

Polycyclic presentation:

2^9|b^2=gh,d^2=gi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,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^c=fi,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, 49594);