#290834

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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

765 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)(9,10,14,19)(11,17,21,16)(12,23,22,24)(13,20,15,18)(25,26,30,33)(27,35,38,32)(28,29,34,31)(36,39,40,37);(1,3,4,8)(2,5,6,7)(9,11,14,21)(10,16,19,17)(12,15,22,13)(18,23,20,24)(25,27,30,38)(26,32,33,35)(28,36,34,40)(29,37,31,39);(9,12,13,21,14,22,15,11)(10,17,18,24,19,16,20,23)(25,28,31,26,30,34,29,33)(27,36,37,35,38,40,39,32);(2,6)(5,7)(9,13,14,15)(10,18,19,20)(11,12,21,22)(16,23,17,24)(25,29,30,31)(26,28,33,34)(27,37,38,39)(32,36,35,40)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fhi,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, 198030);