#8722293

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

4426 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)(3,5)(4,7)(6,8)(9,10,15,14)(11,16,19,20)(12,17,21,23)(13,18,22,24)(25,26)(27,29)(28,36)(30,32)(31,38)(33,35)(34,37)(39,40);(2,3)(6,7)(9,11)(10,16)(12,13)(14,20)(15,19)(17,18)(21,22)(23,24)(25,27,30,35)(26,29,32,33)(28,34,38,40)(31,39,36,37);(1,2,5,3)(4,6,8,7)(11,19)(12,21)(16,20)(17,23)(25,26,30,32)(27,33,35,29)(28,31,38,36)(34,37,40,39);(1,3,5,2)(4,7,8,6)(9,12)(10,17)(11,13)(14,23)(15,21)(16,18)(19,22)(20,24)(25,26)(27,33)(28,31)(29,35)(30,32)(34,37)(36,38)(39,40);(1,4,5,8)(2,6,3,7)(9,13)(10,18)(11,12)(14,24)(15,22)(16,17)(19,21)(20,23)(25,28)(26,31)(27,34)(29,39)(30,38)(32,36)(33,37)(35,40);(9,14,15,10)(11,20,19,16)(12,17,21,23)(13,18,22,24)(25,29,30,33)(26,27,32,35)(28,37,38,39)(31,40,36,34)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8629489);