#8930507

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

4275 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)(11,18)(12,15)(13,21)(14,16)(17,22)(19,23)(20,24)(25,26,32,30)(27,35,39,33)(28,29,34,31)(36,37,40,38);(3,7)(6,8)(9,11,14,12)(10,15,16,18)(13,19,22,20)(17,23,21,24)(25,27,32,39)(26,33,30,35)(28,40,34,36)(29,37,31,38);(1,2)(3,6)(4,5)(7,8)(10,16)(13,22)(15,18)(19,20)(25,28)(26,31)(27,36)(29,30)(32,34)(33,37)(35,38)(39,40);(9,10)(11,15)(12,18)(13,17)(14,16)(19,23)(20,24)(21,22)(25,29,32,31)(26,28,30,34)(27,37,39,38)(33,40,35,36);(1,3,4,7)(2,6,5,8)(9,12,14,11)(10,15,16,18)(13,20,22,19)(17,23,21,24)(25,30,32,26)(27,33,39,35)(28,29,34,31)(36,37,40,38);(1,2)(3,6)(4,5)(7,8)(9,13,14,22)(10,17,16,21)(11,19,12,20)(15,23,18,24)(25,31,32,29)(26,34,30,28)(27,38,39,37)(33,36,35,40)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=i,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, 8837703);