#9020032

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	#59810	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4358 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)(3,7)(4,5)(6,8)(9,10,14,12)(11,17,18,15)(13,20,21,16)(19,22,24,23)(25,26,30,33)(27,31,34,29)(28,32,36,40)(35,39,37,38);(3,8)(6,7)(10,12)(15,17)(16,20)(22,23)(25,27)(26,31)(28,35)(29,33)(30,34)(32,39)(36,37)(38,40);(1,3,5,8)(2,6,4,7)(10,12)(11,18)(13,21)(22,23)(27,34)(28,36)(29,31)(32,40);(3,8)(6,7)(9,11,14,18)(10,15,12,17)(13,19,21,24)(16,22,20,23)(25,27)(26,31)(28,37)(29,33)(30,34)(32,38)(35,36)(39,40);(1,3)(2,7)(4,6)(5,8)(9,12,14,10)(11,15,18,17)(13,20,21,16)(19,22,24,23)(25,28)(26,32)(27,35)(29,38)(30,36)(31,39)(33,40)(34,37);(1,4,5,2)(3,6,8,7)(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,29,30,31)(26,27,33,34)(28,38,36,39)(32,35,40,37)

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8927228);