#8919201

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

4617 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:

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

Polycyclic presentation:

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

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, 8826397);