#7715833

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

4566 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)(11,18)(12,14)(13,16)(15,17)(19,24)(20,23)(21,22)(28,36)(32,37)(35,39)(38,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,16)(12,20)(13,15)(14,21)(17,18)(19,22)(23,24)(26,30)(27,33)(32,37)(35,39);(1,3,5,4)(2,7,6,8)(10,17)(11,13)(12,19)(21,23)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(9,12,15,19)(10,14,17,24)(11,20,13,22)(16,21,18,23);(1,4,5,3)(2,8,6,7)(9,13,15,11)(10,18,17,16)(12,22,19,20)(14,23,24,21)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,39)(37,40);(9,14)(10,19)(11,21)(12,17)(13,23)(15,24)(16,22)(18,20)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40)

Polycyclic presentation:

2^9|c^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=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, 7623029);