#10464592

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

4750 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=g,d^2=g,e^2=h,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, 10371788);