#8186966

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4292 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=gi,e^2=h,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, 8094162);