#9057765

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	#53914	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4247 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=g,e^2=g,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, 8964961);