#8106226

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4356 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)(25,26,30,34)(27,31,32,28)(29,37,38,33)(35,36,40,39);(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)(26,31)(28,34)(29,35)(30,32)(33,39)(36,37)(38,40);(2,5)(6,8)(10,16)(15,19)(17,22)(18,23)(25,28,30,31)(26,32,34,27)(29,36,38,39)(33,40,37,35);(1,2)(3,6)(4,5)(7,8)(9,10)(11,19)(12,17)(13,18)(14,16)(15,21)(20,22)(23,24)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,40)(38,39);(9,12,14,20)(10,17,16,22)(11,13,21,24)(15,23,19,18)(25,29,30,38)(26,33,34,37)(27,35,32,40)(28,36,31,39);(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)(26,28)(29,35)(30,32)(31,34)(33,36)(37,39)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,e^2=h,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, 8013422);