#8838450

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

4542 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)(9,10,14,16)(11,18,21,15)(12,22,17,13)(19,24,23,20)(25,26,30,33)(27,29,34,31)(28,32,38,40)(35,37,36,39);(1,3,5,8)(2,6,7,4)(9,11,14,21)(10,15,16,18)(12,19,17,23)(13,24,22,20)(27,34)(29,31)(35,36)(37,39);(1,2)(3,4)(5,7)(6,8)(10,16)(12,17)(15,18)(19,23)(25,27)(26,31)(28,36)(29,33)(30,34)(32,37)(35,38)(39,40);(1,4)(2,3)(5,6)(7,8)(9,12,14,17)(10,13,16,22)(11,19,21,23)(15,24,18,20)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,40)(34,36);(1,3)(2,4)(5,8)(6,7)(9,11)(10,15)(12,23)(13,20)(14,21)(16,18)(17,19)(22,24)(25,27)(26,31)(28,36)(29,33)(30,34)(32,37)(35,38)(39,40);(1,4)(2,3)(5,6)(7,8)(9,13)(10,17)(11,20)(12,16)(14,22)(15,19)(18,23)(21,24)(25,29,30,31)(26,27,33,34)(28,37,38,39)(32,35,40,36)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8745646);