#7983237

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

4530 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7890433);