#8424278

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

4696 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)(11,17)(12,19)(13,16)(14,15)(18,21)(20,24)(22,23)(25,26)(27,32)(28,29)(30,38)(31,34)(33,39)(35,36)(37,40);(1,2)(3,8)(4,6)(5,7)(10,15)(12,17)(16,23)(20,21)(25,27)(26,32)(28,34)(29,31)(30,35)(33,40)(36,38)(37,39);(1,3,5,6)(2,4,7,8)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,23)(22,24)(25,28)(26,27)(29,34)(30,36)(31,32)(33,35)(37,38)(39,40);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,21)(14,17)(15,19)(16,24)(18,23)(20,22)(25,29,31,27)(26,28,34,32)(30,37,39,35)(33,36,38,40);(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,23)(22,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,40)(34,38);(1,4)(2,6)(3,7)(5,8)(9,13,14,22)(10,16,15,23)(11,18,19,24)(12,20,17,21)(25,28)(26,29)(27,34)(30,36)(31,32)(33,37)(35,38)(39,40)

Polycyclic presentation:

2^9|c^2=ghi,d^2=gh,f^2=g,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, 8331474);