#241090

order	512
nilpotency class	3
isoclinism	isoclinic class of #241087
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1141 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)(4,6)(7,8)(9,10)(11,21)(12,25)(13,18)(14,19)(15,20)(16,23)(17,28)(22,34)(24,32)(26,31)(27,36)(29,35)(30,33)(37,38)(39,40);(1,3)(2,5)(4,7)(6,8)(9,11,15,24)(10,16,20,33)(12,26,29,22)(13,30,14,23)(17,34,36,31)(18,21,19,32)(25,39,35,37)(27,38,28,40);(5,8)(9,12,13,27,15,29,14,28)(10,17,19,35,20,36,18,25)(11,22,23,38,24,26,30,40)(16,31,21,37,33,34,32,39);(9,13,15,14)(10,18,20,19)(11,23,24,30)(12,27,29,28)(16,32,33,21)(17,25,36,35)(22,38,26,40)(31,39,34,37)

Polycyclic presentation:

2^9|b^2=h,c^2=e,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gi,c^b=cg,d^b=dh,e^b=eh,f^b=fi,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, 148286);