#407421

order	512
nilpotency class	3
isoclinism	isoclinic class of #393940
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

960 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 314617);