#8395261

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

4484 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)(11,18)(12,13)(14,15)(16,21)(17,22)(19,20)(23,24)(25,26,31,33)(27,29,28,35)(30,32,38,40)(34,37,36,39);(1,2)(3,8)(4,7)(5,6)(10,15)(12,22)(16,18)(19,23)(25,26)(27,29)(28,35)(30,32)(31,33)(34,37)(36,39)(38,40);(2,6)(4,8)(10,15)(13,17)(16,18)(20,24)(25,27,31,28)(26,29,33,35)(30,36,38,34)(32,39,40,37);(2,6)(3,7)(9,11,14,21)(10,16,15,18)(12,23,22,19)(13,20,17,24)(25,28,31,27)(26,29,33,35)(30,36,38,34)(32,37,40,39);(1,3,5,7)(2,4,6,8)(9,12)(10,17)(11,19)(13,15)(14,22)(16,20)(18,24)(21,23)(25,29)(26,28)(27,33)(30,37)(31,35)(32,36)(34,40)(38,39);(1,4)(2,7)(3,6)(5,8)(9,13,14,17)(10,12,15,22)(11,20,21,24)(16,23,18,19)(25,30,31,38)(26,32,33,40)(27,34,28,36)(29,37,35,39)

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8302457);