#8734258

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

4655 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

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, 8641454);