#8754855

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

4384 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)(9,10,14,15)(11,18,13,16)(12,20,22,17)(19,23,21,24)(25,26,29,30)(27,33,35,31)(28,32,36,37)(34,39,40,38);(3,7)(5,8)(10,15)(16,18)(17,20)(23,24)(26,30)(28,36)(31,33)(34,40);(1,2)(3,5)(4,6)(7,8)(9,11,14,13)(10,16,15,18)(12,21,22,19)(17,24,20,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(9,12,14,22)(10,17,15,20)(11,19,13,21)(16,23,18,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,3)(2,5)(4,7)(6,8)(9,13,14,11)(10,18,15,16)(12,21,22,19)(17,24,20,23)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39);(1,2)(3,5)(4,6)(7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,22)(21,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39)

Polycyclic presentation:

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

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, 8662051);