#8195953

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

4414 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)(11,18)(12,20)(13,15)(14,16)(17,22)(19,23)(21,24)(25,26,30,32)(27,31,34,28)(29,37,38,33)(35,36,40,39);(1,2,5,4)(3,7,8,6)(9,11,14,13)(10,15,16,18)(12,19,22,21)(17,23,20,24)(25,27)(26,31)(28,32)(29,35)(30,34)(33,39)(36,37)(38,40);(2,4)(6,7)(10,16)(15,18)(17,20)(23,24)(26,32)(27,34)(33,37)(35,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,19)(13,21)(14,22)(15,23)(16,20)(18,24)(25,28,30,31)(26,27,32,34)(29,36,38,39)(33,35,37,40);(9,13,14,11)(10,18,16,15)(12,21,22,19)(17,24,20,23)(25,29,30,38)(26,33,32,37)(27,35,34,40)(28,36,31,39);(1,4,5,2)(3,7,8,6)(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,22)(21,24)(25,27)(26,31)(28,32)(29,35)(30,34)(33,39)(36,37)(38,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=gh,e^2=h,f^2=ghi,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, 8103149);