#153299

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	#30496	C₂
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

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

normal subgroups only

This group has 407 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:

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

Polycyclic presentation:

2^9|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, 60495);