#8807321

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

4401 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:

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

Polycyclic presentation:

2^9|c^2=gh,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8714517);