#8093788

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

4316 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=gh,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000984);