#135532

order	512
nilpotency class	4
isoclinism	isoclinic class of #135531
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8747
commutator	C₂²×Q₈	C₂²×C₄
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	#3486	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

451 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 451 conjugacy classes of subgroups — too many to draw. Only its 60 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=dh,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=di,f^b=fgi,h^b=hi,d^c=di,e^c=egi,f^c=fhi,g^c=gi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42728);