#8599816

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

4403 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^2=g,e^2=h,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, 8507012);