#290819

order	512
nilpotency class	3
isoclinism	isoclinic class of #268139
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	#45985	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

806 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)(4,6)(5,8)(9,10)(11,16)(12,22)(13,17)(14,18)(15,19)(20,24)(21,23)(25,26)(27,35)(28,29)(30,34)(31,33)(32,38)(36,39)(37,40);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,13)(14,20)(15,21)(17,22)(18,24)(19,23)(25,27,30,38)(26,32,34,35)(28,36,33,40)(29,37,31,39);(9,12,14,11,15,20,13,21)(10,16,17,22,19,23,18,24)(25,28,29,26,30,33,31,34)(27,36,39,35,38,40,37,32);(2,6)(5,7)(9,13,15,14)(10,17,19,18)(11,12,21,20)(16,22,23,24)(25,29,30,31)(26,33,34,28)(27,37,38,39)(32,40,35,36)

Polycyclic presentation:

2^9|b^2=e,c^2=fi,d^2=h,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,g^c=gh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198015);