#8197092

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×#540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4321 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=ghi,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104288);