#10216505

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

4285 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=ghi,e^2=ghi,f^2=ghi,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, 10123701);