#10457956

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

4537 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,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, 10365152);