#8583460

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4312 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)(3,7)(4,8)(5,6)(25,26,30,33)(27,31,35,29)(28,36,39,32)(34,38,37,40);(1,3,5,8)(2,4,6,7)(9,10)(11,16)(12,20)(13,18)(14,15)(17,22)(19,23)(21,24)(25,27)(26,31)(28,37)(29,33)(30,35)(32,38)(34,39)(36,40);(1,4,5,7)(2,3,6,8)(10,15)(12,22)(16,24)(18,19)(25,28,30,39)(26,32,33,36)(27,34,35,37)(29,38,31,40);(1,2,5,6)(3,7,8,4)(9,11)(10,16)(12,19)(13,17)(14,21)(15,24)(18,22)(20,23)(25,29)(26,27)(28,38)(30,31)(32,34)(33,35)(36,37)(39,40);(1,4)(2,3)(5,7)(6,8)(9,12)(10,17)(11,19)(13,16)(14,22)(15,20)(18,21)(23,24)(25,28)(26,32)(27,34)(29,38)(30,39)(31,40)(33,36)(35,37);(1,2)(3,4)(5,6)(7,8)(9,13)(10,18)(11,20)(12,16)(14,23)(15,19)(17,21)(22,24)(25,27)(26,31)(28,34)(29,33)(30,35)(32,40)(36,38)(37,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,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, 8490656);