#415775

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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 ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

874 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)(3,8,4,6)(17,18,20,22)(19,21,23,24)(25,26,29,33,30,34,28,32)(27,35,38,31,36,39,37,40);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,23)(22,24)(25,27,30,36)(26,31,34,40)(28,38,29,37)(32,39,33,35);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(19,23)(21,24)(26,32)(28,29)(31,39)(33,34)(35,40)(37,38);(9,11,12,15)(10,14,13,16)(18,22)(21,24)(27,36)(31,40)(35,39)(37,38)

Polycyclic presentation:

2^9|a^2=ehi,b^2=efi,c^2=ef,d^2=gh,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,e^b=ei,f^b=fi,e^c=ei,f^c=fi

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, 322971);