#105069

order	512
nilpotency class	3
isoclinism	isoclinic class of #105041
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#394
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

435 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 435 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=d,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,c^b=cf,d^b=dg,e^b=ei,f^b=fg,e^c=egh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12265);