#8446027

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

4489 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,e^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, 8353223);