#8112530

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

4528 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8019726);