#8826224

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

4371 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)(4,8)(6,7)(9,10,14,16)(11,15,17,24)(12,20,22,13)(18,19,21,23)(25,26,30,31)(27,32,33,28)(29,37,38,34)(35,36,40,39);(2,7)(3,8)(9,11)(10,15)(12,18)(13,23)(14,17)(16,24)(19,20)(21,22)(25,26,30,31)(27,32,33,28)(29,37,38,34)(35,36,40,39);(1,3,6,8)(2,4,7,5)(11,17)(15,24)(18,21)(19,23)(26,31)(27,33)(34,37)(35,40);(1,4,6,5)(2,3,7,8)(9,11)(10,15)(12,21)(13,19)(14,17)(16,24)(18,22)(20,23)(25,27)(26,32)(28,31)(29,35)(30,33)(34,39)(36,37)(38,40);(1,2)(3,4)(5,8)(6,7)(9,12)(10,13)(11,18)(14,22)(15,23)(16,20)(17,21)(19,24)(25,28,30,32)(26,33,31,27)(29,36,38,39)(34,40,37,35);(1,5,6,4)(2,8,7,3)(9,13)(10,12)(11,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,29)(26,34)(27,35)(28,36)(30,38)(31,37)(32,39)(33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8733420);