#7979802

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4219 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7886998);