#8917771

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

4592 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)(9,10,14,16)(11,15,19,24)(12,17,21,23)(13,18,20,22)(25,26,30,31)(27,28,33,32)(29,37,39,34)(35,40,38,36);(3,7)(5,8)(9,11,14,19)(10,15,16,24)(12,20,21,13)(17,22,23,18)(26,31)(28,32)(34,37)(36,40);(1,2)(3,5)(4,6)(7,8)(10,16)(11,19)(12,21)(18,22)(25,27,30,33)(26,32,31,28)(29,38,39,35)(34,36,37,40);(9,12)(10,17)(11,20)(13,19)(14,21)(15,22)(16,23)(18,24)(25,27)(26,32)(28,31)(29,35)(30,33)(34,40)(36,37)(38,39);(1,3)(2,5)(4,7)(6,8)(9,13,14,20)(10,18,16,22)(11,21,19,12)(15,23,24,17)(25,28,30,32)(26,33,31,27)(29,36,39,40)(34,38,37,35);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,24)(20,22)(21,23)(25,29)(26,34)(27,35)(28,36)(30,39)(31,37)(32,40)(33,38)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8824967);