#7718456

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4243 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=h,d^2=h,e^2=i,f^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, 7625652);