#213457

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

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, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

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, 120653);