#8460731

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×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4284 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)(3,7)(4,6)(5,8)(10,16)(11,12)(18,22)(19,21)(25,26,30,33)(27,29,32,39)(28,35,38,31)(34,40,36,37);(1,3)(2,5)(4,8)(6,7)(9,10,15,16)(11,17,12,20)(13,18,23,22)(14,19,24,21)(25,27,30,32)(26,29,33,39)(28,36,38,34)(31,40,35,37);(9,11,15,12)(10,17,16,20)(13,19,23,21)(14,18,24,22)(25,28,30,38)(26,31,33,35)(27,34,32,36)(29,37,39,40);(1,2,4,6)(3,5,8,7)(9,12,15,11)(10,17,16,20)(13,21,23,19)(14,18,24,22)(25,27)(26,29)(28,34)(30,32)(31,37)(33,39)(35,40)(36,38);(1,3)(2,5)(4,8)(6,7)(9,13,15,23)(10,18,16,22)(11,19,12,21)(14,17,24,20);(1,2)(3,5)(4,6)(7,8)(9,14,15,24)(10,19,16,21)(11,18,12,22)(13,17,23,20)(25,29)(26,32)(27,33)(28,37)(30,39)(31,36)(34,35)(38,40)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8367927);