#470984

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

784 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=efhi,c^2=efi,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi

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, 378180);