#9003063

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

4500 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)(11,19)(12,17)(13,23)(14,15)(16,20)(18,24)(21,22)(25,26)(27,31)(28,32)(29,37)(30,34)(33,38)(35,40)(36,39);(2,6)(4,7)(10,15)(12,22)(16,19)(18,23)(25,27,30,28)(26,31,34,32)(29,35,38,36)(33,39,37,40);(1,2)(3,4)(5,6)(7,8)(10,15)(11,20)(12,22)(13,24)(25,26,30,34)(27,32,28,31)(29,37,38,33)(35,39,36,40);(1,3)(2,4)(5,8)(6,7)(9,11)(10,16)(12,23)(13,21)(14,20)(15,19)(17,24)(18,22)(25,28,30,27)(26,32,34,31)(29,36,38,35)(33,40,37,39);(1,4)(2,3)(5,7)(6,8)(9,12,14,22)(10,17,15,21)(11,18,20,23)(13,19,24,16)(25,26)(27,32)(28,31)(29,33)(30,34)(35,40)(36,39)(37,38);(1,3)(2,4)(5,8)(6,7)(9,13)(10,18)(11,21)(12,19)(14,24)(15,23)(16,22)(17,20)(25,29,30,38)(26,33,34,37)(27,35,28,36)(31,39,32,40)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8910259);