#240705

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	#60799	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

971 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)(11,20)(12,25)(13,18)(14,30)(15,19)(16,23)(17,29)(21,35)(22,32)(24,34)(26,40)(27,39)(28,33)(31,37)(36,38);(3,8)(5,6)(9,11)(10,16)(12,26)(13,23)(14,22)(15,24)(17,31)(18,34)(19,32)(20,30)(21,29)(25,38)(27,36)(28,37)(33,40)(35,39);(2,5)(3,8)(6,7)(9,12,14,28,15,29,13,27)(10,17,18,33,19,25,30,39)(11,21,23,37,24,26,22,36)(16,31,20,35,32,38,34,40);(9,13,15,14)(10,18,19,30)(11,22,24,23)(12,27,29,28)(16,20,32,34)(17,33,25,39)(21,36,26,37)(31,35,38,40)

Polycyclic presentation:

2^9|c^2=e,d^2=h,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, 147901);