#8946461

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

4486 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:

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

Polycyclic presentation:

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

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, 8853657);