#134581

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

404 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=f,b^2=di,c^2=i,e^2=hi,f^2=ghi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,h^a=hi,e^c=eh,f^c=fghi,g^c=gi,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, 41777);