#8730650

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

4338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

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, 8637846);