#7711559

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
prime radical	C₂×#538	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4394 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)(11,17)(12,20)(13,16)(14,19)(15,21)(18,22)(23,24)(25,26)(27,35)(28,38)(29,33)(30,34)(31,37)(32,39)(36,40);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24)(25,27,30,37)(26,31,34,35)(28,36,39,29)(32,40,38,33);(1,3,5,4)(2,7,6,8)(9,10,13,16)(11,17,19,14)(12,15,21,20)(18,23,24,22)(25,26,30,34)(27,31,37,35)(28,32,39,38)(29,33,36,40);(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,37)(36,40)(38,39);(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,28)(26,32)(27,36)(29,37)(30,39)(31,40)(33,35)(34,38);(1,4,5,3)(2,8,6,7)(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)(25,29,30,36)(26,33,34,40)(27,28,37,39)(31,32,35,38)

Polycyclic presentation:

2^9|b^2=hi,c^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7618755);