#8587190

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	#35352	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4639 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,e^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, 8494386);