#8342257

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

4540 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=gh,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8249453);