#8195900

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

4305 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=g,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, 8103096);