#218033

order	512
nilpotency class	3
isoclinism	isoclinic class of #32712
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	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

528 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 125229);