#8126994

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	#52005	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4516 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8034190);