#153298

order	512
nilpotency class	5
isoclinism	isoclinic class of #153297
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13422
commutator, Frattini, cosocle	#532	C₂³
prime radical	#3018	C₂²
LCS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

389 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 389 conjugacy classes of subgroups — too many to draw. Only its 46 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=d,d^2=i,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ehi,f^a=fgh,g^a=gi,c^b=cf,d^b=di,e^b=eg,g^b=gi,h^b=hi,e^c=eh,f^c=fi,g^c=gh,f^e=fh,g^f=gi

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, 60494);