#9545656

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

4768 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)(9,10,14,15)(11,19,20,16)(12,17,23,22)(13,21,24,18)(25,26)(27,30)(28,31)(29,34)(32,37)(33,36)(35,39)(38,40);(2,3)(6,7)(10,15)(12,23)(16,19)(18,21)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(1,3,5,2)(4,7,8,6)(10,15)(11,20)(12,23)(13,24)(26,27)(31,33)(32,34)(38,39);(1,4)(2,6)(3,7)(5,8)(9,11)(10,16)(12,18)(13,22)(14,20)(15,19)(17,24)(21,23)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(9,12)(10,17)(11,21)(13,16)(14,23)(15,22)(18,20)(19,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(9,13)(10,18)(11,22)(12,16)(14,24)(15,21)(17,20)(19,23)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gh,d^2=ghi,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9452852);