#8114416

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

4404 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=gi,e^2=gi,f^2=h,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, 8021612);