#144630

order	512
nilpotency class	4
isoclinism	isoclinic class of #144629
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2029
commutator, Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	#3486	C₂²
LCS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

507 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 507 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=di,b^2=d,e^2=i,f^2=i,b^a=bd,c^a=ce,e^a=egi,f^a=fghi,c^b=cf,e^b=eh,f^b=fgi,d^c=dg,e^c=ei,f^c=fi,h^c=hi,f^e=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, 51826);