#134410

order	512
nilpotency class	4
isoclinism	isoclinic class of #129554
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

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 ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

344 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 110 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)(5,6,10,11,9,8,7,12)(13,14)(15,23,24,39)(16,19)(17,29)(18,32)(20,25,34,41)(21,31)(22,28)(26,37,33,40)(27,43)(30,38,42,36)(35,44);(1,2,3,4)(5,7,9,10)(6,11,8,12)(14,19)(20,34)(21,35)(22,32)(23,39)(29,43)(36,38)(37,40);(5,8)(6,9)(7,11)(10,12)(13,15,16,24)(14,20,19,34)(17,30,27,42)(18,33,28,26)(21,37,35,40)(22,38,32,36)(23,29,39,43)(25,31,41,44)

Polycyclic presentation:

2^9|a^2=f,b^2=di,c^2=i,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=ch,e^c=egi,f^c=fi,g^c=gi

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, 41606);