#8599642

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	#40963	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4368 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8506838);