#8583461

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	#40537	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4596 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=g,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, 8490657);