#142816

order	512
nilpotency class	4
isoclinism	isoclinic class of #142236
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

731 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 731 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 128

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

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

Polycyclic presentation:

2^9|d^2=gi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,f^b=fg,g^b=gi,h^b=hi,d^c=dh,f^c=fg,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50012);