#349616

order	512
nilpotency class	3
isoclinism	isoclinic class of #346740
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

858 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=i,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,d^b=dh,f^c=fh,g^c=gi,f^d=fi,g^d=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 256812);