#316365

order	512
nilpotency class	3
isoclinism	isoclinic class of #262731
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

732 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)(3,7)(4,6)(5,8)(9,10,14,17)(11,18,19,24)(12,16,13,21)(15,22,23,20)(25,26,28,33)(27,35,37,31)(29,32,30,34)(36,39,38,40);(1,3,4,8)(2,5,6,7)(11,19)(15,23)(18,24)(20,22)(25,27,29,36,28,37,30,38)(26,31,32,40,33,35,34,39);(9,11)(10,15)(12,20)(13,22)(14,19)(16,24)(17,23)(18,21)(26,32)(27,36)(29,30)(31,35)(33,34)(37,38);(2,6)(5,7)(9,12,14,13)(10,16,17,21)(11,20,19,22)(15,24,23,18)

Polycyclic presentation:

2^9|a^2=h,b^2=eghi,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cg,f^c=fh,g^c=gi

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, 223561);