#7892194

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

4513 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=i,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7799390);