#10451672

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

4346 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)(10,14)(12,20)(15,17)(18,21)(25,26,31,36)(27,28,33,32)(29,34,38,40)(30,35,39,37);(10,14)(15,17)(16,22)(23,24)(25,27,31,33)(26,32,36,28)(29,37,38,35)(30,34,39,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(25,28,31,32)(26,33,36,27)(29,30,38,39)(34,35,40,37);(3,7)(6,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,22)(21,23)(25,27)(26,28)(29,37)(30,34)(31,33)(32,36)(35,38)(39,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,21)(13,19)(14,17)(16,24)(18,20)(22,23)(25,29)(26,34)(27,37)(28,30)(31,38)(32,39)(33,35)(36,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,18)(13,20)(14,22)(15,23)(17,24)(19,21)(25,30)(26,35)(27,34)(28,38)(29,32)(31,39)(33,40)(36,37)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

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, 10358868);