#144650

order	512
nilpotency class	4
isoclinism	isoclinic class of #144650
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10392
commutator, Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂³×D₄	Q₈
LCS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

445 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 445 conjugacy classes of subgroups — too many to draw. Only its 46 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:

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

Polycyclic presentation:

2^9|a^2=dg,b^2=d,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=di,e^a=egi,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fg,h^b=hi,d^c=dg,e^c=ei,e^d=ei

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51846);