#7779426

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

4447 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,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, 7686622);