#230951

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

581 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=dei,d^c=di,f^c=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 138147);