#8436752

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4386 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)(3,7)(4,6)(5,8)(10,15)(11,20)(18,21)(19,24)(25,26)(27,35)(28,32)(29,33)(30,34)(31,36)(37,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,10,14,15)(11,16,20,17)(12,18,22,21)(13,19,23,24)(27,36)(28,38)(31,35)(32,39);(9,11)(10,16)(12,19)(13,21)(14,20)(15,17)(18,23)(22,24)(25,27,30,36)(26,31,34,35)(28,37,38,29)(32,33,39,40);(2,6)(5,7)(9,11)(10,17)(12,19)(13,18)(14,20)(15,16)(21,23)(22,24)(25,28)(26,32)(27,37)(29,36)(30,38)(31,33)(34,39)(35,40);(1,3)(2,5)(4,8)(6,7)(9,12,14,22)(10,18,15,21)(11,19,20,24)(13,16,23,17)(25,29,30,37)(26,33,34,40)(27,28,36,38)(31,39,35,32);(1,2)(3,5)(4,6)(7,8)(9,13)(10,19)(11,21)(12,16)(14,23)(15,24)(17,22)(18,20)(25,26)(27,31)(28,39)(29,33)(30,34)(32,38)(35,36)(37,40)

Polycyclic presentation:

2^9|b^2=hi,c^2=i,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

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, 8343948);