#418086

order	512
nilpotency class	3
isoclinism	isoclinic class of #417597
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1468 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325282);