#8359171

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

4492 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8266367);