#128231

order	512
nilpotency class	3
isoclinism	isoclinic class of #108504
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#376
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1745
prime radical	C₂³	#432
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

226 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=d,d^2=gi,e^2=h,h^2=i,b^a=bd,c^a=ch,d^a=di,f^a=fgi,e^b=eg,e^c=ei

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, 35427);