#8112478

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	#42997	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4616 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8019674);