#8106095

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	#35261	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4504 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)(11,12)(13,21)(14,15)(16,18)(17,22)(19,24)(20,23)(26,30)(27,33)(28,35)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,24)(22,23)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(2,5)(6,8)(10,15)(11,16)(17,21)(19,23)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,18)(12,16)(13,17)(14,15)(19,24)(20,23)(21,22)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(9,12,14,18)(10,16,15,11)(13,20,22,24)(17,23,21,19)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,3)(2,6)(4,7)(5,8)(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)

Polycyclic presentation:

2^9|b^2=g,c^2=g,e^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

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, 8013291);