#144048

order	512
nilpotency class	4
isoclinism	isoclinic class of #143943
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#459
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

412 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2)(3,4)(5,6,8,11)(7,9,12,10)(14,20)(15,26)(16,31)(17,34)(18,35)(21,40)(22,32)(23,24)(25,37)(27,42)(28,38)(30,36)(39,41)(43,44);(2,4)(13,14,19,25)(15,27,30,39)(16,32,35,23)(17,22,34,24)(18,37,31,20)(21,33,43,29)(26,44,36,40)(28,41,38,42);(6,9)(7,12)(10,11)(13,15,19,30)(14,21,25,43)(16,33,35,29)(17,36,34,26)(18,38,31,28)(20,39,37,27)(22,42,24,41)(23,44,32,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=i,d^2=gi,e^2=h,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,g^a=gi,c^b=cf,d^b=dg,f^b=fgi,g^b=gi,d^c=dg,e^c=eh,f^c=fgi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51244);