#8115860

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4362 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)(9,10,14,16)(11,18,21,15)(12,22,17,13)(19,20,23,24)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,16,18)(12,19,17,23)(13,20,22,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(2,5)(6,8)(10,16)(12,17)(15,18)(19,23)(28,36)(31,38)(34,40)(37,39);(9,11)(10,15)(12,19)(13,20)(14,21)(16,18)(17,23)(22,24)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,2)(3,6)(4,5)(7,8)(9,12,14,17)(10,13,16,22)(11,19,21,23)(15,20,18,24)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39);(1,3)(2,6)(4,7)(5,8)(9,13)(10,17)(11,20)(12,16)(14,22)(15,23)(18,19)(21,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40)

Polycyclic presentation:

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

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, 8023056);