#8182078

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

4621 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=g,e^2=g,f^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, 8089274);