#8699353

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	#45057	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4580 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)(12,20)(13,15)(14,17)(16,22)(19,23)(21,24)(25,26,31,37)(27,32,39,40)(28,34,38,33)(29,30,36,35);(3,7)(6,8)(9,11,14,13)(10,15,17,18)(12,21,22,19)(16,24,20,23)(25,27)(26,32)(28,35)(29,34)(30,38)(31,39)(33,36)(37,40);(1,2)(3,6)(4,5)(7,8)(25,28)(26,33)(27,30)(29,32)(31,38)(34,37)(35,39)(36,40);(1,3)(2,6)(4,7)(5,8)(9,12,14,22)(10,16,17,20)(11,19,13,21)(15,23,18,24)(25,28)(26,34)(27,35)(29,32)(30,39)(31,38)(33,37)(36,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,18)(14,17)(19,23)(20,22)(21,24)(25,29)(26,35)(27,34)(28,32)(30,37)(31,36)(33,39)(38,40);(9,13,14,11)(10,15,17,18)(12,19,22,21)(16,24,20,23)(25,30,31,35)(26,36,37,29)(27,38,39,28)(32,33,40,34)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=g,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, 8606549);