#8664365

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4533 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)(4,8)(5,6)(9,10,16,15)(11,17,14,21)(12,22,13,18)(19,24,20,23)(26,32)(27,28)(29,37)(38,39);(1,3)(2,4)(5,8)(6,7)(9,11,16,14)(10,17,15,21)(12,20,13,19)(18,24,22,23)(25,26,31,32)(27,30,28,35)(29,33,37,40)(34,39,36,38);(9,12,16,13)(10,18,15,22)(11,19,14,20)(17,23,21,24)(25,27,31,28)(26,30,32,35)(29,34,37,36)(33,39,40,38);(2,6)(4,7)(9,13,16,12)(10,18,15,22)(11,20,14,19)(17,23,21,24)(25,28,31,27)(26,30,32,35)(29,36,37,34)(33,39,40,38);(1,3)(2,7)(4,6)(5,8)(9,14,16,11)(10,17,15,21)(12,20,13,19)(18,23,22,24)(25,29,31,37)(26,33,32,40)(27,34,28,36)(30,39,35,38);(1,4,5,7)(2,8,6,3)(9,15,16,10)(11,21,14,17)(12,22,13,18)(19,24,20,23)(25,30)(26,28)(27,32)(29,38)(31,35)(33,34)(36,40)(37,39)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8571561);