#8990321

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
prime radical	#41982	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4419 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)(3,8)(4,7)(5,6)(25,26,30,32)(27,33,34,29)(28,31,36,39)(35,40,37,38);(3,4)(7,8)(9,10,14,11)(12,15,19,17)(13,16,21,18)(20,22,24,23)(25,26)(27,29)(28,31)(30,32)(33,34)(35,38)(36,39)(37,40);(1,3)(2,7)(4,6)(5,8)(10,11)(12,19)(13,21)(22,23)(27,34)(28,36)(29,33)(31,39);(3,4)(7,8)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,27)(26,29)(28,37)(30,34)(31,40)(32,33)(35,36)(38,39);(1,4,6,3)(2,7,5,8)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,28)(26,31)(27,35)(29,38)(30,36)(32,39)(33,40)(34,37);(1,5,6,2)(3,7,4,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,27)(28,38)(30,33)(31,35)(32,34)(36,40)(37,39)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=hi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8897517);