#134577

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	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

420 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=di,e^2=hi,f^2=gh,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, 41773);