#8689791

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

4373 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8596987);