#7824350

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

4399 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7731546);