#10368418

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
prime radical	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4097 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10275614);