#9063938

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

4473 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8971134);