#8449301

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

4560 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8356497);