#8837067

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

4546 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:

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

Polycyclic presentation:

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

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, 8744263);