#8195789

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

4212 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=g,d^2=h,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, 8102985);