#10086632

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

4482 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9993828);