#8106078

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

4640 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)(10,14)(11,17)(16,21)(18,23)(26,31)(27,36)(28,38)(29,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26)(27,32)(28,34)(29,35)(30,31)(33,36)(37,39)(38,40);(2,5)(6,8)(11,17)(15,20)(18,23)(22,24)(25,27,30,36)(26,32,31,33)(28,37,38,35)(29,34,39,40);(1,2)(3,6)(4,5)(7,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,27)(26,33)(28,37)(29,34)(30,36)(31,32)(35,38)(39,40);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,28,30,38)(26,34,31,40)(27,37,36,35)(29,32,39,33);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,29)(26,35)(27,34)(28,32)(30,39)(31,37)(33,38)(36,40)

Polycyclic presentation:

2^9|c^2=h,d^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, 8013274);