#7989742

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
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4830 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^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, 7896938);