#8630880

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
prime radical	#37825	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4445 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)(3,4)(5,8)(6,7)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,26)(27,34)(28,38)(29,37)(30,32)(31,33)(35,39)(36,40);(1,3)(2,4)(5,6)(7,8)(10,14)(15,17)(16,21)(22,23)(25,27,32,37)(26,29,30,34)(28,35,31,36)(33,39,38,40);(1,4)(2,5)(3,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28,32,31)(26,33,30,38)(27,35,37,36)(29,39,34,40);(1,3,6,5)(2,4,7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,29)(26,27)(28,39)(30,37)(31,40)(32,34)(33,35)(36,38);(1,5,6,3)(2,4,7,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,30,32,26)(27,29,37,34)(28,33,31,38)(35,40,36,39);(1,4)(2,3)(5,7)(6,8)(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,31,32,28)(26,33,30,38)(27,36,37,35)(29,39,34,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=hi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8538076);