#10424601

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

4356 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)(3,5)(4,6)(7,8)(9,10,15,20)(11,16,22,24)(12,13,18,17)(14,23,21,19)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,38)(39,40);(2,3)(6,7)(9,11,15,22)(10,16,20,24)(12,21,18,14)(13,19,17,23)(27,31)(28,35)(34,39)(36,40);(9,12)(10,17)(11,21)(13,20)(14,22)(15,18)(16,23)(19,24)(25,27,30,31)(26,28,33,35)(29,34,38,39)(32,36,37,40);(2,3)(4,8)(9,12)(10,13)(11,21)(14,22)(15,18)(16,19)(17,20)(23,24)(25,26)(27,28)(29,37)(30,33)(31,35)(32,38)(34,40)(36,39);(1,3,5,2)(4,7,8,6)(9,13)(10,18)(11,19)(12,20)(14,16)(15,17)(21,24)(22,23)(25,28)(26,31)(27,33)(29,36)(30,35)(32,39)(34,37)(38,40);(1,4,5,8)(2,6,3,7)(9,14,15,21)(10,19,20,23)(11,12,22,18)(13,16,17,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,e^2=g,f^2=gh,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, 10331797);