#9062272

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4290 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8969468);