#8594807

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

4434 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)(11,19)(12,22)(13,20)(14,16)(15,21)(17,23)(18,24)(25,26)(27,33)(28,38)(29,34)(30,32)(31,35)(36,40)(37,39);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,18)(13,22)(14,21)(16,19)(17,24)(20,23)(25,27)(26,33)(28,36)(29,35)(30,39)(31,34)(32,37)(38,40);(2,5)(6,8)(10,16)(13,24)(15,19)(17,22)(25,28,32,29)(26,34,30,38)(27,35,37,36)(31,33,40,39);(3,7)(6,8)(9,12,14,23)(10,17,16,22)(11,20,21,18)(13,19,24,15)(25,29,32,28)(26,34,30,38)(27,36,37,35)(31,33,40,39);(1,2)(3,6)(4,5)(7,8)(9,13)(10,18)(11,17)(12,19)(14,24)(15,23)(16,20)(21,22)(25,30,32,26)(27,33,37,39)(28,34,29,38)(31,36,40,35);(1,3)(2,6)(4,7)(5,8)(9,11)(10,19)(12,18)(13,17)(14,21)(15,16)(20,23)(22,24)(25,31,32,40)(26,35,30,36)(27,34,37,38)(28,33,29,39)

Polycyclic presentation:

2^9|b^2=ghi,c^2=gh,d^2=h,e^2=gh,f^2=gh,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, 8502003);