#8818994

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

4539 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gi,d^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8726190);