#290864

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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

714 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)(9,10,15,19)(11,16,21,23)(12,22,20,24)(13,17,14,18)(25,26)(27,32)(28,35)(29,33)(30,39)(31,34)(36,37)(38,40);(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)(26,32)(28,33)(29,35)(30,40)(31,36)(34,37)(38,39);(9,12,14,11,15,20,13,21)(10,16,17,22,19,23,18,24)(25,28,30,32,31,38,29,37)(26,27,33,35,34,36,39,40);(2,6)(5,7)(9,13,15,14)(10,17,19,18)(11,12,21,20)(16,22,23,24)(25,29,31,30)(26,33,34,39)(27,35,36,40)(28,37,38,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=f,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, 198060);