#8107442

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

4524 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

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, 8014638);