#10069103

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

4476 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)(3,7)(4,5)(6,8)(9,10)(11,14)(12,15)(13,18)(16,21)(17,20)(19,23)(22,24)(25,26)(27,34)(28,32)(29,33)(30,35)(31,37)(36,40)(38,39);(2,5)(6,7)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,27)(26,31)(28,36)(29,38)(30,37)(32,39)(33,40)(34,35);(2,5)(6,7)(10,11)(15,17)(16,18)(22,23)(25,28,30,29)(26,32,35,33)(27,36,37,38)(31,39,34,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,29,30,28)(26,33,35,32)(27,36,37,38)(31,39,34,40);(1,2)(3,7)(4,5)(6,8)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,27)(26,34)(28,38)(29,36)(30,37)(31,35)(32,39)(33,40);(1,3)(2,6)(4,8)(5,7)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,26)(27,31)(28,32)(29,33)(30,35)(34,37)(36,39)(38,40)

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=ghi,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

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, 9976299);