#9054496

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

4693 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)(3,7)(9,10,14,15)(11,12,16,19)(13,17,21,22)(18,20,23,24)(25,26,31,28)(27,29,33,32)(30,36,38,34)(35,39,40,37);(1,2,6,5)(3,8,7,4)(10,15)(12,19)(17,22)(20,24)(25,27)(26,32)(28,29)(30,35)(31,33)(34,39)(36,37)(38,40);(1,3)(2,4)(5,8)(6,7)(11,16)(12,19)(13,21)(17,22)(26,28)(29,32)(30,38)(35,40);(1,2,6,5)(3,4,7,8)(9,11)(10,12)(13,18)(14,16)(15,19)(17,20)(21,23)(22,24)(25,28,31,26)(27,29,33,32)(30,36,38,34)(35,37,40,39);(1,4,6,8)(2,7,5,3)(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,22)(21,24)(25,29)(26,33)(27,28)(30,37)(31,32)(34,40)(35,36)(38,39);(1,5,6,2)(3,8,7,4)(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,20,19,24)(25,30,31,38)(26,34,28,36)(27,35,33,40)(29,37,32,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8961692);