#8509680

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×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4229 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8416876);