#470735

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

980 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:

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

Polycyclic presentation:

2^9|a^2=egi,b^2=h,c^2=fi,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,d^c=de

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 377931);