#8630858

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

4651 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)(3,4)(5,8)(6,7)(9,10,14,17)(11,18,13,15)(12,16,21,24)(19,23,20,22)(26,32)(27,28)(29,37)(39,40);(1,3)(2,4)(5,6)(7,8)(9,11,14,13)(10,15,17,18)(12,20,21,19)(16,23,24,22)(25,26)(27,33)(28,30)(29,38)(31,32)(34,37)(35,39)(36,40);(1,4)(2,5)(3,7)(6,8)(12,21)(16,24)(19,20)(22,23)(25,27,31,28)(26,33,32,30)(29,36,37,35)(34,39,38,40);(1,3,6,5)(2,4,7,8)(9,10)(11,15)(12,16)(13,18)(14,17)(19,22)(20,23)(21,24)(25,28,31,27)(26,33,32,30)(29,36,37,35)(34,40,38,39);(1,5,6,3)(2,4,7,8)(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,24)(18,23)(25,29)(26,34)(27,35)(28,36)(30,39)(31,37)(32,38)(33,40);(1,4)(2,3)(5,7)(6,8)(9,13,14,11)(10,15,17,18)(12,20,21,19)(16,22,24,23)(25,30,31,33)(26,28,32,27)(29,39,37,40)(34,36,38,35)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=hi,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8538054);