#10422909

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	#48776	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4382 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)(11,12)(13,16)(14,17)(15,19)(18,20)(21,22)(23,24)(26,31)(28,35)(33,37)(36,40);(11,15)(12,19)(18,23)(20,24)(25,26)(27,28)(29,33)(30,31)(32,35)(34,36)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,15)(10,12,17,19)(13,18,22,23)(16,20,21,24)(25,27,30,32)(26,28,31,35)(29,34,38,39)(33,36,37,40);(3,7)(6,8)(9,10)(11,12)(13,21)(14,17)(15,19)(16,22)(18,24)(20,23)(25,26)(27,28)(29,37)(30,31)(32,35)(33,38)(34,40)(36,39);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,17)(13,20)(14,19)(16,23)(18,21)(22,24)(25,28)(26,32)(27,31)(29,36)(30,35)(33,39)(34,37)(38,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,21)(19,24)(25,29,30,38)(26,33,31,37)(27,34,32,39)(28,36,35,40)

Polycyclic presentation:

2^9|c^2=gi,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, 10330105);