#8462048

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4571 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8369244);