#8196699

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4189 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)(3,4)(11,17)(12,20)(15,18)(16,21)(25,26,30,34)(27,32,28,31)(29,33,38,37)(35,40,36,39);(1,2,6,5)(3,8,4,7)(9,10,14,13)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,40)(37,38);(1,3,6,4)(2,7,5,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,26)(27,31)(28,32)(29,37)(30,34)(33,38)(35,40)(36,39);(1,4,6,3)(2,8,5,7)(25,27,30,28)(26,31,34,32)(29,35,38,36)(33,39,37,40);(1,5,6,2)(3,8,4,7)(9,12,14,20)(10,16,13,21)(11,19,17,23)(15,22,18,24)(25,28,30,27)(26,32,34,31)(29,36,38,35)(33,40,37,39);(9,13,14,10)(11,15,17,18)(12,21,20,16)(19,22,23,24)(25,29,30,38)(26,33,34,37)(27,35,28,36)(31,39,32,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=ghi,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8103895);