#406986

order	512
nilpotency class	3
isoclinism	isoclinic class of #393944
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

860 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,15,16,13)(17,18)(19,21)(20,22)(23,24)(25,26,30,34)(27,31,37,40)(28,32,29,33)(35,38,36,39);(1,3,5,4)(2,6,8,7)(17,19,20,23)(18,21,22,24)(25,27)(26,31)(28,36)(29,35)(30,37)(32,39)(33,38)(34,40);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(27,35)(28,29)(31,38)(32,33)(36,37)(39,40);(1,4,5,3)(2,6,8,7)(18,22)(21,24)(25,28,30,29)(26,32,34,33)(27,36,37,35)(31,39,40,38)

Polycyclic presentation:

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

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, 314182);