#8758058

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

4551 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)(11,14)(12,20)(13,16)(15,22)(17,19)(18,24)(21,23)(25,26,31,36)(27,34,29,32)(28,38,37,33)(30,35,39,40);(1,2)(3,7)(4,5)(6,8)(9,11,13,19)(10,14,16,17)(12,21,22,18)(15,24,20,23)(25,27,31,29)(26,32,36,34)(28,39,37,30)(33,40,38,35);(2,5)(6,7)(9,10,13,16)(11,17,19,14)(12,15,22,20)(18,24,21,23)(25,28)(26,33)(27,30)(29,39)(31,37)(32,35)(34,40)(36,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,23)(20,22)(21,24)(25,29,31,27)(26,34,36,32)(28,39,37,30)(33,40,38,35);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,19)(15,23)(16,17)(20,24)(21,22)(25,30,31,39)(26,35,36,40)(27,37,29,28)(32,38,34,33);(1,3)(2,6)(4,8)(5,7)(9,12)(10,15)(11,18)(13,22)(14,23)(16,20)(17,24)(19,21)(25,26)(27,32)(28,33)(29,34)(30,35)(31,36)(37,38)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=g,e^2=g,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, 8665254);