#10409716

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

4474 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10316912);