#134587

order	512
nilpotency class	4
isoclinism	isoclinic class of #124662
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 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

326 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)(6,9)(7,11)(10,12)(13,14,18,23,19,24,30,36,16,20,31,37,32,38,17,22)(15,25,28,35,29,41,27,42,26,43,33,21,34,39,44,40);(1,2,3,4)(14,20)(21,35)(22,36)(23,37)(24,38)(25,43)(39,41)(40,42);(5,6,8,10)(7,12,11,9)(13,15)(14,21)(16,26)(17,33)(18,27)(19,34)(20,35)(22,41)(23,25)(24,42)(28,30)(29,32)(31,44)(36,39)(37,43)(38,40)

Polycyclic presentation:

2^9|a^2=f,b^2=di,c^2=g,e^2=hi,f^2=gh,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, 41783);