#10480691

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

4520 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)(11,16)(12,14)(13,20)(15,19)(17,21)(18,22)(23,24)(25,26)(27,32)(28,30)(29,36)(31,35)(33,38)(34,39)(37,40);(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,23)(21,24)(25,26)(27,32)(28,30)(29,33)(31,35)(34,40)(36,38)(37,39);(1,2)(3,6)(4,5)(7,8)(9,10,14,12)(11,16,19,15)(13,17,21,20)(18,23,24,22)(25,27)(26,31)(28,32)(29,34)(30,35)(33,39)(36,40)(37,38);(3,7)(6,8)(9,12,14,10)(11,16,19,15)(13,20,21,17)(18,23,24,22)(25,28,30,26)(27,32,35,31)(29,36,38,33)(34,40,37,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,27)(26,32)(28,31)(29,37)(30,35)(33,39)(34,38)(36,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,40)(35,37)

Polycyclic presentation:

2^9|c^2=g,d^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,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, 10387887);