#10171844

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

4302 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)(9,10,14,13)(11,17,19,15)(12,20,21,16)(18,22,24,23)(26,30)(27,33)(32,35)(34,36);(1,3,5,8)(2,6,7,4)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,26)(27,31)(28,35)(29,30)(32,37)(33,38)(34,40)(36,39);(1,3)(2,4)(5,8)(6,7)(25,27,29,33)(26,31,30,38)(28,34,37,36)(32,39,35,40);(1,4)(2,3)(5,6)(7,8)(9,12,14,21)(10,16,13,20)(11,18,19,24)(15,22,17,23)(25,27)(26,31)(28,36)(29,33)(30,38)(32,40)(34,37)(35,39);(9,13,14,10)(11,17,19,15)(12,16,21,20)(18,22,24,23)(25,28,29,37)(26,32,30,35)(27,34,33,36)(31,39,38,40);(9,11)(10,17)(12,18)(13,15)(14,19)(16,23)(20,22)(21,24)(25,26)(27,31)(28,35)(29,30)(32,37)(33,38)(34,40)(36,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=ghi,e^2=gi,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, 10079040);