#8371584

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	#37284	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4423 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:

(2,5)(6,8)(25,26,31,34)(27,29,28,32)(30,33,38,40)(35,37,36,39);(3,7)(6,8)(9,10)(11,16)(12,13)(14,15)(17,22)(18,24)(19,21)(20,23)(25,27,31,28)(26,32,34,29)(30,35,38,36)(33,39,40,37);(1,2)(3,6)(4,5)(7,8)(10,15)(13,17)(16,23)(19,24)(25,27,31,28)(26,29,34,32)(30,36,38,35)(33,39,40,37);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,21)(13,19)(14,20)(15,23)(17,24)(18,22)(25,28,31,27)(26,29,34,32)(30,36,38,35)(33,37,40,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,18)(13,15)(14,22)(16,19)(20,21)(23,24)(25,29)(26,28)(27,34)(30,37)(31,32)(33,36)(35,40)(38,39);(9,13,14,17)(10,12,15,22)(11,19,20,24)(16,21,23,18)(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=gh,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, 8278780);