#9054528

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

4387 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)(9,10,15,12)(11,13,17,16)(14,20,22,18)(19,23,24,21)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(1,2,6,5)(3,8,7,4)(9,11)(10,16)(12,13)(14,19)(15,17)(18,23)(20,21)(22,24)(29,37)(32,36)(34,40)(38,39);(1,3)(2,4)(5,8)(6,7)(10,12)(13,16)(14,22)(19,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,2,6,5)(3,4,7,8)(9,12,15,10)(11,13,17,16)(14,20,22,18)(19,21,24,23)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(1,4,6,8)(2,7,5,3)(9,13)(10,17)(11,12)(14,21)(15,16)(18,24)(19,20)(22,23)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40);(1,5,6,2)(3,8,7,4)(9,14,15,22)(10,18,12,20)(11,19,17,24)(13,21,16,23)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40)

Polycyclic presentation:

2^9|a^2=hi,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, 8961724);