#9054389

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	#52435	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4117 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=g,e^2=ghi,f^2=h,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, 8961585);