#8395271

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

4508 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)(11,20)(12,14)(13,23)(15,17)(16,24)(18,22)(19,21)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(1,2)(3,8)(4,7)(5,6)(9,11,15,24)(10,16,17,20)(12,21,22,13)(14,23,18,19)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(2,6)(4,8)(10,17)(14,18)(16,20)(19,23)(25,26,30,28)(27,31,35,33)(29,36,37,32)(34,39,40,38);(2,6)(3,7)(9,12)(10,18)(11,21)(13,24)(14,17)(15,22)(16,19)(20,23)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,3,5,7)(2,4,6,8)(9,13,15,21)(10,19,17,23)(11,22,24,12)(14,16,18,20)(25,28,30,26)(27,31,35,33)(29,36,37,32)(34,38,40,39);(1,4)(2,7)(3,6)(5,8)(9,14,15,18)(10,12,17,22)(11,23,24,19)(13,16,21,20)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

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, 8302467);