#153301

order	512
nilpotency class	5
isoclinism	isoclinic class of #153119
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2199
commutator, Frattini, cosocle	#532	C₂³
socle	C₂	#13424
prime radical	#3018	C₂²
LCS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 336 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=d,d^2=i,e^2=h,g^2=i,b^a=bd,c^a=ce,e^a=eh,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

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60497);