#153186

order	512
nilpotency class	5
isoclinism	isoclinic class of #153185
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13348
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

486 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 50 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)(3,5)(4,14,18,25)(6,9)(7,23)(8,16,12,21)(10,22)(11,19)(13,24)(15,28,26,17)(20,27)(29,30,32,31);(1,3,5,2)(4,15,18,26)(6,9,19,11)(7,13,20,10)(8,17,12,28)(14,29,25,32)(16,30,21,31)(22,27,24,23);(1,4)(2,8)(3,12)(5,18)(6,21)(7,17)(9,25)(10,26)(11,14)(13,15)(16,19)(20,28)(22,29)(23,30)(24,32)(27,31)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=h,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fghi,g^a=gi,c^b=cf,d^b=dh,e^b=ei,g^b=gh,d^c=dg,e^c=ei,g^c=gi,h^c=hi,f^d=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, 60382);