#8108307

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

4672 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:

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

Polycyclic presentation:

2^9|c^2=g,d^2=gh,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=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, 8015503);