#8930516

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

4185 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)(9,10,15,12)(11,18,13,16)(14,20,22,17)(19,23,21,24)(26,30)(27,33)(28,36)(37,40);(3,7)(6,8)(9,11,15,13)(10,16,12,18)(14,19,22,21)(17,23,20,24)(25,26,31,30)(27,34,33,29)(28,32,36,38)(35,40,39,37);(1,2)(3,6)(4,5)(7,8)(10,12)(14,22)(16,18)(19,21)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,39)(38,40);(9,12,15,10)(11,18,13,16)(14,20,22,17)(19,24,21,23)(25,28)(26,32)(27,35)(29,37)(30,38)(31,36)(33,39)(34,40);(1,3,4,7)(2,6,5,8)(9,13,15,11)(10,16,12,18)(14,21,22,19)(17,23,20,24)(25,29,31,34)(26,33,30,27)(28,37,36,40)(32,39,38,35);(1,2)(3,6)(4,5)(7,8)(9,14,15,22)(10,17,12,20)(11,19,13,21)(16,23,18,24)(25,30,31,26)(27,34,33,29)(28,38,36,32)(35,40,39,37)

Polycyclic presentation:

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

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, 8837712);