#8936026

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

4504 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,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, 8843222);