#302160

order	512
nilpotency class	3
isoclinism	isoclinic class of #269019
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	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

844 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 209356);