#8096082

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

4539 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8003278);