#7764867

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

4613 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)(11,12)(13,17)(14,16)(15,19)(18,20)(21,23)(22,24)(25,26,30,31)(27,28,33,36)(29,32,37,38)(34,35,39,40);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,23)(21,24)(25,26)(27,28)(29,32)(30,31)(33,36)(34,35)(37,38)(39,40);(10,16)(12,15)(17,23)(20,22)(26,31)(27,33)(29,37)(35,40);(1,3,5,4)(2,7,6,8)(9,12,14,15)(10,11,16,19)(13,20,21,22)(17,18,23,24)(25,27,30,33)(26,28,31,36)(29,34,37,39)(32,35,38,40);(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,23)(21,24)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,40)(38,39);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,17,16,23)(11,18,19,24)(12,20,15,22)(25,29,30,37)(26,32,31,38)(27,34,33,39)(28,35,36,40)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=g,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, 7672063);