#8182075

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

4563 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,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, 8089271);