#8509824

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4490 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)(12,15)(13,21)(14,17)(16,22)(19,23)(20,24)(25,26,30,33)(27,34,29,31)(28,32,37,39)(35,40,36,38);(1,2)(3,6)(4,5)(7,8)(11,12)(15,18)(19,20)(23,24)(25,27,30,29)(26,31,33,34)(28,35,37,36)(32,38,39,40);(2,5)(6,8)(9,11,14,12)(10,15,17,18)(13,20,22,19)(16,24,21,23)(25,28,30,37)(26,32,33,39)(27,35,29,36)(31,38,34,40);(3,7)(6,8)(9,12,14,11)(10,15,17,18)(13,20,22,19)(16,23,21,24)(25,26)(27,31)(28,32)(29,34)(30,33)(35,38)(36,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,21)(18,24)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,18)(13,16)(14,17)(19,23)(20,24)(21,22)(25,29,30,27)(26,31,33,34)(28,36,37,35)(32,38,39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=h,f^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, 8417020);