#8288648

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

4680 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8195844);