#8934895

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
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4392 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8842091);