#8947353

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

4262 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8854549);