#471274

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
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 ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

960 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)(11,14)(12,15)(13,16)(17,18,20,22)(19,21,23,24)(25,26,31,35)(27,36,28,39)(29,34,30,33)(32,38,40,37);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(17,19)(18,21)(20,23)(22,24);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,16,14)(19,23)(21,24)(25,27,30,38,31,28,29,37)(26,32,34,39,35,40,33,36);(3,7)(6,8)(9,12,13,11)(10,15,16,14)(25,28,30,37,31,27,29,38)(26,32,34,39,35,40,33,36)

Polycyclic presentation:

2^9|a^2=gi,b^2=fh,c^2=efi,d^2=fi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^c=de

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378470);