#10422679

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

4292 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10329875);