#10422927

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

4485 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gi,e^2=g,f^2=g,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, 10330123);