#8994062

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

4589 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=i,d^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8901258);