#8756980

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=gh,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,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, 8664176);