#8437836

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	#34990	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4632 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)(9,10,14,18)(11,19,21,15)(12,22,20,16)(13,17,23,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(3,7)(6,8)(9,11,14,21)(10,15,18,19)(12,13,20,23)(16,17,22,24)(27,34)(30,31)(35,40)(37,38);(1,2)(3,6)(4,5)(7,8)(25,27)(26,30)(28,35)(29,34)(31,33)(32,37)(36,40)(38,39);(1,3)(2,6)(4,7)(5,8)(9,12)(10,16)(11,13)(14,20)(15,17)(18,22)(19,24)(21,23)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,22)(23,24)(25,28,29,36)(26,32,33,39)(27,35,34,40)(30,37,31,38);(9,13)(10,17)(11,20)(12,21)(14,23)(15,22)(16,19)(18,24)(25,26)(27,30)(28,32)(29,33)(31,34)(35,37)(36,39)(38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8345032);