#136202

order	512
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

411 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 411 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=ghi,d^2=gi,e^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,h^a=hi,d^b=dg,e^b=eh,g^b=gi,h^b=hi,d^c=dh,e^c=egi,g^c=gi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43398);