#8114356

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

4384 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8021552);