#8904618

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

4624 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=h,f^2=gh,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, 8811814);