#8679889

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×#548	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4344 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8587085);