#8726619

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4159 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=i,d^2=h,e^2=gi,f^2=gi,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, 8633815);