#8701709

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4200 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=g,e^2=gi,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, 8608905);