#8155024

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

4725 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8062220);