#8739289

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×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3756 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,9)(3,8,12,7)(4,13,14,5)(10,11,16,15)(18,20)(23,26)(24,28)(30,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,3,6,12)(2,7,9,8)(4,10,14,16)(5,11,13,15)(19,25)(23,26)(27,31)(30,32)(34,38)(39,41)(40,45)(46,47);(1,3,6,12)(2,7,9,8)(4,10,14,16)(5,11,13,15)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,3)(2,8)(4,10)(5,15)(6,12)(7,9)(11,13)(14,16)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,4,6,14)(2,5,9,13)(3,10,12,16)(7,11,8,15)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30);(1,5)(2,4)(3,11)(6,13)(7,10)(8,16)(9,14)(12,15)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=h,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8646485);