#7806501

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

4184 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)(11,15)(12,20)(13,17)(14,18)(16,22)(19,24)(21,23)(25,26,31,34)(27,28,32,29)(30,33,38,40)(35,36,39,37);(1,3,4,8)(2,5,6,7)(9,11,14,13)(10,15,18,17)(12,19,22,21)(16,23,20,24)(27,32)(28,29)(30,38)(33,40);(9,10,14,18)(11,15,13,17)(12,16,22,20)(19,23,21,24)(25,27)(26,28)(29,34)(30,39)(31,32)(33,37)(35,38)(36,40);(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,22)(21,24)(25,28,31,29)(26,32,34,27)(30,36,38,37)(33,39,40,35);(9,12,14,22)(10,16,18,20)(11,19,13,21)(15,23,17,24)(25,29,31,28)(26,27,34,32)(30,37,38,36)(33,35,40,39);(9,13,14,11)(10,17,18,15)(12,21,22,19)(16,24,20,23)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=i,e^2=hi,f^2=h,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, 7713697);