#8699113

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4320 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=gh,f^2=gi,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, 8606309);