#134353

order	512
nilpotency class	3
isoclinism	isoclinic class of #7606
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	C₂×C₂²⋊C₈
prime radical	C₂²×D₄	C₂×C₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

426 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)(7,9)(8,11)(10,12)(13,14,16,21,18,22,27,34,17,20,26,32,28,33,38,42)(15,19,24,30,25,31,37,41,23,29,35,39,36,40,43,44);(1,2,3,4)(13,15)(14,19)(16,24)(17,23)(18,25)(20,29)(21,30)(22,31)(26,35)(27,37)(28,36)(32,39)(33,40)(34,41)(38,43)(42,44);(6,9)(7,12)(10,11)(14,20)(15,23)(16,26)(22,33)(25,36)(27,38)(30,39)(41,44)

Polycyclic presentation:

2^9|a^2=f,b^2=d,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,c^b=ci,e^c=egi,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41549);