#9062285

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	#60044	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4360 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=gi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=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, 8969481);