#9054426

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

4409 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=g,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,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, 8961622);