#8823822

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4120 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=hi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8731018);