#9057800

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	#47889	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4350 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=gh,f^2=gi,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, 8964996);