#240835

order	512
nilpotency class	3
isoclinism	isoclinic class of #240831
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:

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

Polycyclic presentation:

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

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, 148031);