#8839507

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

4855 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=hi,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, 8746703);