#8095321

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4308 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)(3,5)(4,7)(6,8)(9,10,15,16)(11,14,12,17)(13,18,22,23)(19,21,20,24)(29,37)(32,36)(34,40)(38,39);(1,3,4,8)(2,5,7,6)(10,16)(11,12)(18,23)(19,20)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(1,3)(2,6)(4,8)(5,7)(9,11,15,12)(10,17,16,14)(13,20,22,19)(18,21,23,24)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(1,2)(3,6)(4,7)(5,8)(9,12,15,11)(10,14,16,17)(13,20,22,19)(18,21,23,24);(9,13,15,22)(10,18,16,23)(11,19,12,20)(14,21,17,24)(25,28,30,26)(27,31,35,33)(29,36,37,32)(34,38,40,39);(9,14)(10,11)(12,16)(13,21)(15,17)(18,19)(20,23)(22,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,d^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8002517);