#8636189

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4263 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)(11,18)(12,20)(13,14)(15,16)(17,23)(19,22)(21,24)(25,26)(27,32)(28,33)(29,31)(30,34)(35,38)(36,40)(37,39);(1,2,4,5)(3,6,7,8)(9,11,15,14)(10,13,16,18)(12,21,23,19)(17,24,20,22)(25,27,31,34)(26,32,29,30)(28,36,39,35)(33,40,37,38);(2,5)(6,8)(10,16)(13,18)(17,20)(22,24)(26,29)(27,34)(33,37)(35,36);(3,7)(6,8)(9,12,15,23)(10,17,16,20)(11,19,14,21)(13,22,18,24)(25,28,31,39)(26,33,29,37)(27,35,34,36)(30,40,32,38);(1,2)(3,6)(4,5)(7,8)(9,13)(10,11)(12,22)(14,16)(15,18)(17,19)(20,21)(23,24)(25,29,31,26)(27,32,34,30)(28,37,39,33)(35,38,36,40);(1,3,4,7)(2,6,5,8)(9,14,15,11)(10,13,16,18)(12,19,23,21)(17,24,20,22)(25,30,31,32)(26,34,29,27)(28,38,39,40)(33,35,37,36)

Polycyclic presentation:

2^9|b^2=gi,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8543385);