#8505445

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

4454 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)(3,5)(4,7)(6,8)(10,15)(11,17)(12,19)(21,24)(25,26,31,28)(27,32,37,35)(29,38,40,33)(30,34,36,39);(2,3)(6,7)(9,10,14,15)(11,13,17,22)(12,20,19,16)(18,24,23,21)(25,27)(26,32)(28,35)(29,30)(31,37)(33,39)(34,38)(36,40);(1,2,5,3)(4,6,8,7)(25,26,31,28)(27,35,37,32)(29,33,40,38)(30,34,36,39);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,18)(14,17)(15,22)(16,21)(19,23)(20,24)(25,28,31,26)(27,35,37,32)(29,38,40,33)(30,34,36,39);(1,4,5,8)(2,6,3,7)(9,12,14,19)(10,16,15,20)(11,18,17,23)(13,21,22,24)(25,29)(26,33)(27,36)(28,38)(30,37)(31,40)(32,34)(35,39);(9,13)(10,17)(11,15)(12,21)(14,22)(16,23)(18,20)(19,24)(25,30,31,36)(26,34,28,39)(27,29,37,40)(32,38,35,33)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8412641);