#8703516

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

4525 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)(12,20,21,16)(18,22,24,23)(25,26)(27,30)(28,38)(29,40)(31,35)(32,37)(33,39)(34,36);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,27)(26,32)(28,29)(30,35)(31,37)(33,40)(34,38)(36,39);(1,3)(2,5)(4,8)(6,7)(12,21)(16,20)(18,24)(22,23)(25,28,31,39)(26,33,35,38)(27,29,37,36)(30,34,32,40);(3,8)(5,6)(9,12,14,21)(10,16,13,20)(11,18,19,24)(15,22,17,23)(25,28)(26,33)(27,36)(29,37)(30,40)(31,39)(32,34)(35,38);(1,2)(3,5)(4,7)(6,8)(9,11)(10,17)(12,18)(13,15)(14,19)(16,23)(20,22)(21,24)(25,29,31,36)(26,34,35,40)(27,28,37,39)(30,33,32,38);(1,3)(2,6)(4,8)(5,7)(9,13,14,10)(11,17,19,15)(12,16,21,20)(18,22,24,23)(25,30,31,32)(26,27,35,37)(28,34,39,40)(29,38,36,33)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8610712);