#8734966

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

4467 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=g,e^2=gh,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, 8642162);