#7862618

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	#39120	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4589 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7769814);