#213456

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=f,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, 120652);