#8475783

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

4690 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=g,d^2=gh,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, 8382979);