#472650

order	512
nilpotency class	3
isoclinism	isoclinic class of #454382
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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

844 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=ghi,c^2=ef,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,g^c=gi

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, 379846);