#8188182

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	#45023	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4480 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:

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

Polycyclic presentation:

2^9|b^2=gi,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8095378);