#259498

order	512
nilpotency class	3
isoclinism	isoclinic class of #246591
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

753 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)(4,5)(6,8)(9,10,12,16)(11,18,20,14)(13,15,22,17)(19,24,21,23);(2,5)(6,7)(9,11)(10,14)(12,20)(13,19)(15,23)(16,18)(17,24)(21,22)(25,26,30,33)(27,34,37,40)(28,32,29,38)(31,35,39,36);(10,15)(11,19)(13,22)(16,17)(20,21)(23,24)(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)(14,18)(15,17)(23,24)(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=hi,c^2=hi,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dhi,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 166694);