#8837068

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

4203 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=hi,d^2=gh,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, 8744264);