#10040937

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
prime radical	#43910	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4490 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=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, 9948133);