#153300

order	512
nilpotency class	5
isoclinism	isoclinic class of #153299
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

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

normal subgroups only

This group has 391 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)(3,13,18,19)(4,5,11,7)(6,20,23,29)(9,17,26,15)(10,24,28,21)(14,31,32,30)(16,25,22,27);(1,3)(2,9)(4,19)(5,17)(6,22)(7,15)(8,18)(10,27)(11,13)(12,26)(14,20)(16,23)(21,31)(24,30)(25,28)(29,32);(1,4,5,12,8,11,7,2)(3,14,15,27,18,32,17,25)(6,20,21,28,23,29,24,10)(9,22,13,30,26,16,19,31)

Polycyclic presentation:

2^9|a^2=i,c^2=d,d^2=i,e^2=hi,f^2=i,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,f^b=fi,g^b=gi,h^b=hi,e^c=eh,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, 60496);