#8743458

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

4605 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)(11,17)(12,15)(13,16)(14,19)(18,24)(20,22)(21,23)(25,26)(27,31)(28,37)(29,33)(30,34)(32,39)(35,36)(38,40);(3,7)(5,8)(9,10,13,16)(11,14,19,17)(12,15,22,20)(18,23,21,24)(25,26)(27,35)(28,32)(29,33)(30,34)(31,36)(37,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,14,19,17)(12,15,22,20)(18,23,21,24)(25,27,30,36)(26,31,34,35)(28,38,39,29)(32,33,37,40);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,20)(13,16)(15,22)(17,19)(18,24)(21,23)(25,28,30,39)(26,32,34,37)(27,29,36,38)(31,40,35,33);(1,2)(3,5)(4,6)(7,8)(9,11,13,19)(10,14,16,17)(12,21,22,18)(15,24,20,23)(25,26)(27,31)(28,32)(29,40)(30,34)(33,38)(35,36)(37,39);(9,12)(10,15)(11,18)(13,22)(14,23)(16,20)(17,24)(19,21)(25,29)(26,33)(27,28)(30,38)(31,37)(32,35)(34,40)(36,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gi,d^2=i,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, 8650654);