#8127470

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	#35301	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4440 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)(3,7)(4,5)(6,8)(13,21)(15,22)(18,24)(20,23)(26,30)(28,35)(31,36)(34,39);(2,5)(6,7)(9,10,14,16)(11,17,19,12)(13,15,21,22)(18,23,24,20)(27,33)(31,36)(34,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,19)(10,12,16,17)(13,18,21,24)(15,20,22,23)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40);(9,13)(10,15)(11,18)(12,20)(14,21)(16,22)(17,23)(19,24)

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=h,e^2=h,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, 8034666);