#238325

order	512
nilpotency class	3
isoclinism	isoclinic class of #238324
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

868 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,f^b=fi,e^c=eh,f^c=fhi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145521);