#10534364

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

4505 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,16)(13,19)(14,17)(18,21)(20,23)(22,24)(25,26,30,35)(27,34,37,31)(28,38,39,32)(29,33,36,40);(1,3)(2,6)(4,7)(5,8)(9,11,14,19)(10,15,17,13)(12,18,22,20)(16,21,24,23)(25,27)(26,31)(28,36)(29,39)(30,37)(32,40)(33,38)(34,35);(9,11)(10,13)(12,18)(14,19)(15,17)(16,23)(20,22)(21,24)(25,28,30,39)(26,32,35,38)(27,36,37,29)(31,40,34,33);(1,2)(3,4)(5,7)(6,8)(9,11)(10,13)(12,20)(14,19)(15,17)(16,21)(18,22)(23,24)(25,29)(26,33)(27,28)(30,36)(31,32)(34,38)(35,40)(37,39);(1,2)(3,6)(4,8)(5,7)(9,12)(10,16)(11,18)(13,23)(14,22)(15,21)(17,24)(19,20)(25,27)(26,34)(28,36)(29,39)(30,37)(31,35)(32,33)(38,40);(1,4,5,6)(2,3,7,8)(9,13)(10,11)(12,21)(14,15)(16,20)(17,19)(18,24)(22,23)(25,26)(27,31)(28,32)(29,33)(30,35)(34,37)(36,40)(38,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,f^2=ghi,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, 10441560);