#8757492

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

4221 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664688);