#10482502

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

4799 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=i,e^2=g,f^2=h,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, 10389698);