#10322610

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

4518 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)(11,12)(13,21)(14,15)(16,23)(17,20)(18,24)(19,22)(25,26)(27,33)(28,30)(29,32)(31,35)(34,39)(36,37)(38,40);(1,2)(3,7)(4,5)(6,8)(9,11,14,17)(10,12,15,20)(13,22,23,18)(16,24,21,19)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(2,5)(6,7)(10,15)(11,17)(16,21)(18,22)(29,37)(32,36)(34,40)(38,39);(2,5)(6,7)(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,22)(23,24)(25,28,30,26)(27,31,35,33)(29,32,37,36)(34,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39);(1,3)(2,6)(4,8)(5,7)(9,13,14,23)(10,16,15,21)(11,18,17,22)(12,19,20,24)(25,27)(26,33)(28,31)(29,34)(30,35)(32,39)(36,38)(37,40)

Polycyclic presentation:

2^9|b^2=gh,d^2=g,e^2=g,f^2=h,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, 10229806);