#9065476

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

4490 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,e^2=g,f^2=h,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, 8972672);