#8664777

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

4651 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,d^2=gh,e^2=g,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, 8571973);