#8942624

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

4456 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=gh,e^2=h,f^2=h,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, 8849820);