#9034404

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

4522 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:

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

Polycyclic presentation:

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

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, 8941600);