#8196376

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	#59318	C₂
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) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=gh,e^2=h,f^2=ghi,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, 8103572);