#7782721

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

4524 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)(11,19)(12,17)(13,22)(14,16)(15,20)(18,23)(21,24)(25,26)(27,37)(28,34)(29,31)(30,35)(32,36)(33,38)(39,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,13)(14,20)(16,19)(17,18)(21,23)(22,24)(25,27,32,38)(26,33,36,37)(28,39,30,31)(29,35,40,34);(10,16)(11,20)(12,21)(18,22)(25,28,32,30)(26,34,36,35)(27,31,38,39)(29,33,40,37);(9,12,14,21)(10,17,16,24)(11,13,20,23)(15,18,19,22)(25,29)(26,31)(27,35)(28,33)(30,37)(32,40)(34,38)(36,39);(1,3,5,4)(2,7,6,8)(9,13)(10,18)(11,12)(14,23)(15,17)(16,22)(19,24)(20,21)(25,30,32,28)(26,35,36,34)(27,31,38,39)(29,33,40,37);(1,4,5,3)(2,8,6,7)(9,10)(11,15)(12,17)(13,18)(14,16)(19,20)(21,24)(22,23)(25,31)(26,29)(27,28)(30,38)(32,39)(33,35)(34,37)(36,40)

Polycyclic presentation:

2^9|b^2=hi,c^2=hi,d^2=h,e^2=gh,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, 7689917);