#110011

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#120
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1436
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

501 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 501 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=hi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,d^b=dh,e^b=ei,d^c=di,e^c=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17207);