#10153060

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

4425 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10060256);