#10042527

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4124 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:

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

Polycyclic presentation:

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

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, 9949723);