#7806299

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4226 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

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, 7713495);