#8651224

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

4351 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)(12,20)(13,14)(15,16)(17,23)(19,22)(21,24)(25,26)(27,33)(28,38)(29,34)(30,32)(31,35)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,14)(10,13,16,18)(12,21,23,19)(17,24,20,22)(25,27,32,35)(26,33,30,31)(28,37,29,36)(34,39,38,40);(2,3)(6,7)(10,16)(13,18)(17,20)(22,24)(25,28,32,29)(26,34,30,38)(27,36,35,37)(31,39,33,40);(4,8)(6,7)(9,12,15,23)(10,17,16,20)(11,19,14,21)(13,22,18,24)(25,29,32,28)(26,34,30,38)(27,37,35,36)(31,39,33,40);(1,3,5,2)(4,7,8,6)(9,13)(10,11)(12,22)(14,16)(15,18)(17,19)(20,21)(23,24)(25,30,32,26)(27,33,35,31)(28,34,29,38)(36,39,37,40);(1,4,5,8)(2,6,3,7)(9,14,15,11)(10,13,16,18)(12,19,23,21)(17,24,20,22)(25,31,32,33)(26,35,30,27)(28,39,29,40)(34,37,38,36)

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=h,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8558420);