#258288

order	512
nilpotency class	3
isoclinism	isoclinic class of #246239
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

847 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)(9,10,13,16)(11,17,19,14)(12,15,20,23)(18,21,24,22);(2,5)(6,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,24)(21,23)(25,26,30,33)(27,34,37,40)(28,32,29,38)(31,35,39,36);(9,12)(10,15)(11,18)(13,20)(14,21)(16,23)(17,22)(19,24)(25,27,30,37)(26,31,33,39)(28,36,29,35)(32,40,38,34);(1,3,4,8)(2,6,5,7)(10,16)(12,20)(14,17)(18,24)(25,28,30,29)(26,32,33,38)(27,35,37,36)(31,34,39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=ehi,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=dhi,g^c=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 165484);