#10320376

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

4263 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gi,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,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, 10227572);