#8093706

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
prime radical	#45454	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4436 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:

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

Polycyclic presentation:

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

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, 8000902);