#8226815

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

4585 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)(11,17,18,16)(12,21,22,13)(19,20,24,23)(25,26,30,32)(27,31,35,40)(28,33,38,37)(29,36,39,34);(1,3,4,7)(2,5,6,8)(10,15)(11,18)(12,22)(20,23)(25,27)(26,31)(28,34)(29,33)(30,35)(32,40)(36,38)(37,39);(1,2)(3,5)(4,6)(7,8)(10,15)(13,21)(16,17)(20,23)(26,32)(27,35)(33,37)(34,36);(1,3)(2,5)(4,7)(6,8)(9,11,14,18)(10,16,15,17)(12,19,22,24)(13,20,21,23)(25,28,30,38)(26,33,32,37)(27,36,35,34)(29,31,39,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,13)(11,19)(14,22)(15,21)(16,20)(17,23)(18,24)(25,29,30,39)(26,34,32,36)(27,37,35,33)(28,31,38,40);(9,13)(10,12)(11,20)(14,21)(15,22)(16,19)(17,24)(18,23)(25,27)(26,31)(28,36)(29,37)(30,35)(32,40)(33,39)(34,38)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=gi,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8134011);