#10324512

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

4549 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)(9,10,14,16)(11,15,19,24)(12,17,21,22)(13,20,23,18)(25,26,30,35)(27,36,37,31)(28,33,38,32)(29,39,40,34);(9,11)(10,15)(12,20)(13,22)(14,19)(16,24)(17,23)(18,21)(25,27,30,37)(26,31,35,36)(28,34,38,39)(29,33,40,32);(1,2)(3,6)(4,5)(7,8)(10,16)(11,19)(17,22)(18,20)(25,28,30,38)(26,32,35,33)(27,34,37,39)(29,36,40,31);(3,7)(6,8)(9,12)(10,17)(11,18)(13,15)(14,21)(16,22)(19,20)(23,24)(25,27)(26,31)(28,34)(29,32)(30,37)(33,40)(35,36)(38,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,24)(20,21)(22,23)(25,28)(26,33)(27,34)(29,36)(30,38)(31,40)(32,35)(37,39);(1,2)(3,6)(4,5)(7,8)(9,13)(10,18)(11,17)(12,15)(14,23)(16,20)(19,22)(21,24)(25,29,30,40)(26,34,35,39)(27,32,37,33)(28,36,38,31)

Polycyclic presentation:

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

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, 10231708);