#8096471

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

4388 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:

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

Polycyclic presentation:

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

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, 8003667);