#8810072

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

4444 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8717268);