#406808

order	512
nilpotency class	3
isoclinism	isoclinic class of #393899
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 314004);