#258123

order	512
nilpotency class	3
isoclinism	isoclinic class of #246159
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

884 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)(3,7)(4,5)(6,8)(9,10)(11,14)(12,15)(13,16)(17,22)(18,21)(19,23)(20,24);(2,5)(6,7)(9,11)(10,14)(12,17)(13,18)(15,22)(16,21)(19,24)(20,23)(25,26,30,33)(27,34,36,39)(28,32,29,37)(31,38,40,35);(9,12)(10,15)(11,17)(13,19)(14,20)(16,23)(18,24)(21,22)(25,27,30,36)(26,31,33,40)(28,35,29,38)(32,39,37,34);(1,3,4,8)(2,6,5,7)(12,19)(15,23)(17,24)(20,22)(25,28,30,29)(26,32,33,37)(27,35,36,38)(31,39,40,34)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dh,g^c=gi

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, 165319);