#240704

order	512
nilpotency class	3
isoclinism	isoclinic class of #240703
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
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

947 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fhi,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, 147900);