#10336068

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

4494 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)(9,10,14,16)(11,17,19,13)(12,15,20,23)(18,24,21,22)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,39)(36,40);(9,11,14,19)(10,13,16,17)(12,18,20,21)(15,22,23,24)(27,28)(29,38)(31,34)(32,37);(1,2)(3,6)(4,5)(7,8)(12,20)(15,23)(18,21)(22,24)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,21)(14,19)(15,24)(16,17)(18,20)(22,23)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,18)(13,22)(14,20)(16,23)(17,24)(19,21)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38);(1,2)(3,6)(4,5)(7,8)(9,13)(10,11)(12,22)(14,17)(15,18)(16,19)(20,24)(21,23)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)

Polycyclic presentation:

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

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, 10243264);