#7989839

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

4276 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7897035);