#8617775

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

4479 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=gh,e^2=hi,f^2=h,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, 8524971);