#8754848

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

4230 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^2=i,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, 8662044);