#8509685

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

4352 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8416881);