#8106003

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	#40562	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4535 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8013199);