#408172

order	512
nilpotency class	3
isoclinism	isoclinic class of #396836
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

749 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)(3,6)(4,8)(5,7)(10,16)(11,21)(12,26)(13,14)(17,22)(19,20)(24,25)(27,28)(30,33)(31,39)(35,37)(36,40);(1,3,5,4)(2,6,7,8)(9,10)(11,22)(12,18)(13,19)(14,16)(15,20)(17,25)(21,32)(23,36)(24,29)(26,33)(27,30)(28,34)(31,38)(35,40)(37,39);(2,7)(6,8)(9,11,13,24,15,25,14,21)(10,17,16,29,20,22,19,32)(12,23,26,37,28,38,27,35)(18,31,30,39,34,36,33,40);(1,4,5,3)(2,8,7,6)(9,12,14,27,15,28,13,26)(10,18,16,30,20,34,19,33)(11,23,21,35,25,38,24,37)(17,31,29,39,22,36,32,40)

Polycyclic presentation:

2^9|b^2=fh,c^2=e,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,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, 315368);