#8936600

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

4448 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)(11,12)(13,21)(14,15)(16,22)(17,20)(18,24)(19,23)(25,26,31,29)(27,28,35,32)(30,37,39,33)(34,40,38,36);(1,2)(3,6)(4,5)(7,8)(9,11,14,17)(10,12,15,20)(13,18,22,23)(16,19,21,24)(25,27,31,35)(26,32,29,28)(30,34,39,38)(33,40,37,36);(2,5)(3,7)(10,15)(11,17)(13,22)(19,24)(25,27,31,35)(26,32,29,28)(30,38,39,34)(33,36,37,40);(3,7)(6,8)(9,12,14,20)(10,11,15,17)(13,19,22,24)(16,18,21,23)(25,28)(26,27)(29,35)(30,36)(31,32)(33,34)(37,38)(39,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,18)(14,17)(15,20)(16,19)(21,24)(22,23)(25,29,31,26)(27,32,35,28)(30,37,39,33)(34,40,38,36);(1,2)(3,6)(4,5)(7,8)(9,13,14,22)(10,16,15,21)(11,18,17,23)(12,19,20,24)(25,30,31,39)(26,33,29,37)(27,34,35,38)(28,36,32,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=gh,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,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, 8843796);