#8424507

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

4517 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)(13,21)(14,17)(16,22)(19,23)(20,24)(26,31)(27,36)(28,35)(38,39);(1,2)(3,8)(4,6)(5,7)(11,12)(15,18)(19,20)(23,24)(25,26)(27,32)(28,34)(29,35)(30,31)(33,36)(37,39)(38,40);(1,3,5,6)(2,4,7,8)(9,11,14,12)(10,15,17,18)(13,19,22,20)(16,23,21,24)(25,27)(26,32)(28,37)(29,38)(30,36)(31,33)(34,39)(35,40);(1,2)(3,4)(5,7)(6,8)(9,12,14,11)(10,15,17,18)(13,20,22,19)(16,23,21,24)(25,27)(26,33)(28,37)(29,39)(30,36)(31,32)(34,38)(35,40);(9,13,14,22)(10,16,17,21)(11,19,12,20)(15,23,18,24)(25,28,30,35)(26,34,31,29)(27,37,36,40)(32,39,33,38);(1,4)(2,6)(3,7)(5,8)(9,10)(11,15)(12,18)(13,16)(14,17)(19,23)(20,24)(21,22)(25,29,30,34)(26,35,31,28)(27,38,36,39)(32,40,33,37)

Polycyclic presentation:

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

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, 8331703);