#8346676

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	C₂×#2635	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4691 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)(11,18)(12,20)(13,17)(14,15)(16,21)(19,23)(22,24)(25,26)(27,32)(28,35)(29,33)(30,31)(34,39)(36,40)(37,38);(1,2)(3,7)(4,5)(6,8)(10,15)(11,17)(12,21)(23,24)(26,31)(32,35)(33,38)(39,40);(2,5)(6,7)(10,15)(13,18)(16,20)(23,24)(25,27,30,28)(26,32,31,35)(29,36,37,34)(33,40,38,39);(3,8)(6,7)(9,11,14,17)(10,13,15,18)(12,22,21,19)(16,24,20,23)(25,28,30,27)(26,32,31,35)(29,36,37,34)(33,39,38,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,16)(11,19)(13,23)(14,21)(15,20)(17,22)(18,24)(25,26)(27,32)(28,35)(29,33)(30,31)(34,39)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(9,13)(10,17)(11,15)(12,23)(14,18)(16,22)(19,20)(21,24)(25,29)(26,33)(27,34)(28,36)(30,37)(31,38)(32,39)(35,40)

Polycyclic presentation:

2^9|c^2=gh,d^2=g,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, 8253872);