#8399493

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	#40198	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4321 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8306689);