#256264

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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

916 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)(3,6)(4,5)(7,8)(9,10,12,15)(11,17,19,14)(13,16,21,24)(18,23,20,22)(27,35)(28,29)(31,39)(32,33)(36,40)(37,38);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,21)(23,24)(25,26,30,34)(27,36,37,39)(28,33,29,32)(31,35,40,38);(1,3,4,7)(2,6,5,8)(11,18)(13,21)(14,22)(16,24)(17,23)(19,20)(25,27,30,37)(26,31,34,40)(28,38,29,35)(32,36,33,39);(2,5)(6,8)(10,15)(14,17)(16,24)(22,23)(27,37)(31,40)(35,38)(36,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^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=dh,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, 163460);