#8689764

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	#59026	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4584 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:

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

Polycyclic presentation:

2^9|c^2=g,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8596960);