#8709054

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	#53425	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4598 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)(9,10)(11,15)(12,18)(13,17)(14,16)(19,22)(20,24)(21,23)(25,26)(27,30)(28,38)(29,40)(31,35)(32,37)(33,39)(34,36);(3,7)(6,8)(9,11,14,18)(10,15,16,12)(13,19,23,24)(17,22,21,20)(25,27)(26,32)(28,29)(30,35)(31,37)(33,40)(34,38)(36,39);(1,2)(3,6)(4,5)(7,8)(10,16)(11,18)(17,21)(19,24)(25,28,31,39)(26,33,35,38)(27,29,37,36)(30,34,32,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,12)(13,21)(14,16)(15,18)(17,23)(19,22)(20,24)(25,28)(26,33)(27,36)(29,37)(30,40)(31,39)(32,34)(35,38);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,22)(14,15)(16,18)(17,24)(19,21)(20,23)(25,29,31,36)(26,34,35,40)(27,28,37,39)(30,33,32,38);(9,13)(10,17)(11,19)(12,20)(14,23)(15,22)(16,21)(18,24)(25,30,31,32)(26,27,35,37)(28,34,39,40)(29,38,36,33)

Polycyclic presentation:

2^9|b^2=i,c^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8616250);