#261041

order	512
nilpotency class	3
isoclinism	isoclinic class of #246799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

685 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,f^c=fh,g^c=ghi

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, 168237);