#8154707

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	#34562	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4596 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)(12,20)(16,21)(18,24)(22,23)(25,26)(27,33)(28,35)(29,37)(30,32)(31,38)(34,36)(39,40);(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,26,30,32)(27,33,35,28)(29,31,38,37)(34,39,40,36);(1,3)(2,5)(4,7)(6,8)(10,14)(15,17)(16,21)(22,23)(25,26,30,32)(27,33,35,28)(29,31,38,37)(34,39,40,36);(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27,30,35)(26,28,32,33)(29,34,38,40)(31,36,37,39);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,40)(38,39);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,31)(27,34)(28,36)(30,38)(32,37)(33,39)(35,40)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^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^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, 8061903);