#8695319

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	#59419	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4445 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8602515);