#8371332

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4344 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,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, 8278528);