#8215459

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
prime radical	#58013	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4207 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:

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

Polycyclic presentation:

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

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, 8122655);