#8159258

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

4321 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8066454);