#8107652

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	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4434 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=hi,e^2=h,f^2=g,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, 8014848);