#8932374

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

4570 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)(3,7)(4,6)(5,8)(9,10,14,12)(11,15,17,19)(13,16,21,20)(18,22,23,24)(25,26)(27,28)(29,37)(30,33)(31,35)(32,38)(34,39)(36,40);(1,2,4,6)(3,5,8,7)(10,12)(15,19)(16,20)(22,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,37)(38,40);(1,2)(3,5)(4,6)(7,8)(11,17)(13,21)(15,19)(16,20)(25,26,30,33)(27,31,35,28)(29,37,38,32)(34,36,40,39);(1,3,4,8)(2,5,6,7)(9,11,14,17)(10,15,12,19)(13,18,21,23)(16,22,20,24)(25,26)(27,31)(28,35)(29,32)(30,33)(34,39)(36,40)(37,38);(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,37,40);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,38)(26,32,33,37)(27,34,35,40)(28,36,31,39)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839570);