#144407

order	512
nilpotency class	4
isoclinism	isoclinic class of #144372
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
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

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

normal subgroups only

This group has 771 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) = 24
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2)(3,5)(4,6)(7,8)(10,15,17,20)(16,22,24,18);(1,3)(2,5)(4,7)(6,8)(9,10)(11,15)(12,20)(13,22)(14,17)(16,19)(18,23)(21,24);(5,8)(9,11,14,12)(10,16,17,24)(13,21,23,19)(15,18,20,22)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,f^2=i,b^a=bd,c^a=ce,f^a=fghi,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dg,f^c=fi,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51603);