#10219796

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	#51131	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4315 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10126992);