#8839073

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

4558 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)(3,7,8,4)(9,10,14,15)(11,18,17,13)(12,20,21,16)(19,23,22,24)(25,26,30,32)(27,31,33,28)(29,37,39,34)(35,36,38,40);(2,6)(4,7)(10,15)(12,21)(13,18)(19,22)(25,27)(26,31)(28,32)(29,38)(30,33)(34,36)(35,39)(37,40);(1,2)(3,4)(5,6)(7,8)(10,15)(11,17)(16,20)(19,22)(26,32)(27,33)(34,37)(35,38);(1,3)(2,4)(5,8)(6,7)(9,11,14,17)(10,13,15,18)(12,22,21,19)(16,24,20,23)(25,27)(26,31)(28,32)(29,35)(30,33)(34,40)(36,37)(38,39);(1,4)(2,3)(5,7)(6,8)(9,12)(10,16)(11,19)(13,23)(14,21)(15,20)(17,22)(18,24)(25,28,30,31)(26,33,32,27)(29,36,39,40)(34,38,37,35);(1,3)(2,4)(5,8)(6,7)(9,13)(10,17)(11,15)(12,23)(14,18)(16,22)(19,20)(21,24)(25,29,30,39)(26,34,32,37)(27,35,33,38)(28,36,31,40)

Polycyclic presentation:

2^9|a^2=h,d^2=hi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8746269);