#134603

order	512
nilpotency class	4
isoclinism	isoclinic class of #124662
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

392 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 118 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dgi,c^2=i,e^2=hi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,e^a=eg,e^c=eh,f^c=fgh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41799);