#153647

order	512
nilpotency class	6
isoclinism	isoclinic class of #153645
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13674
commutator, Frattini, cosocle	#494	C₂³
LCS	G ▷ #494 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #494 ◁ G
LECS	G ▷ #494 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

267 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 30 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)(3,7)(4,8)(5,11)(6,13)(9,16)(10,18)(12,21)(14,23)(15,25)(17,29)(19,30)(20,28)(22,26)(24,31)(27,32);(2,7,13,9)(4,17)(5,21)(8,23,10,24)(11,25,12,26)(14,19)(15,22)(18,20)(27,29,30,28)(31,32);(2,8)(4,18)(7,23)(9,24)(10,13)(11,29)(12,28)(14,31)(17,20)(19,32)(25,30)(26,27)

Polycyclic presentation:

2^9|b^2=hi,d^2=i,e^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fg,g^a=gi,h^a=hi,e^b=efgh,g^b=gh,d^c=df,e^c=ei,f^c=fhi,g^c=ghi,e^d=egh,f^d=fhi,g^d=gi,f^e=fhi,g^e=gi

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60843);