#8424881

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

4377 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=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, 8332077);