#8301440

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

4624 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8208636);