#8898056

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

4636 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805252);