#8933177

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

4550 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=h,e^2=g,f^2=g,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, 8840373);