#7764860

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

4611 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,f^2=g,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, 7672056);