#8904766

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

4396 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)(9,10,14,18)(11,15,19,22)(12,16,21,24)(13,17,23,20)(25,26)(27,32)(28,35)(29,31)(30,37)(33,39)(34,36)(38,40);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,20)(13,16)(14,19)(17,21)(18,22)(23,24)(25,27,31,35)(26,28,29,32)(30,34,39,40)(33,36,37,38);(1,2)(3,5)(4,6)(7,8)(11,19)(12,21)(15,22)(16,24)(25,28,31,32)(26,27,29,35)(30,38,39,36)(33,40,37,34);(9,12)(10,16)(11,20)(13,15)(14,21)(17,19)(18,24)(22,23)(25,29,31,26)(27,32,35,28)(30,37,39,33)(34,38,40,36);(1,3)(2,5)(4,7)(6,8)(9,13)(10,17)(11,16)(12,22)(14,23)(15,21)(18,20)(19,24)(25,27)(26,32)(28,29)(30,34)(31,35)(33,38)(36,37)(39,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,20)(13,16)(14,19)(17,21)(18,22)(23,24)(25,30,31,39)(26,33,29,37)(27,34,35,40)(28,36,32,38)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=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, 8811962);