#133603

order	512
nilpotency class	4
isoclinism	isoclinic class of #133341
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂²×Q₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂³	#379
LCS	G ▷ C₂²×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

522 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40799);