#8186916

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4029 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)(3,10)(4,7)(5,13)(6,9)(8,14)(11,15)(12,16)(18,23)(19,26)(20,28)(29,31)(33,34,39,38)(35,42,37,40)(36,44,46,41)(43,47,45,48);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,18,22,23)(19,21,26,24)(20,25,28,30)(27,29,32,31)(33,35,39,37)(34,40,38,42)(36,43,46,45)(41,47,44,48);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(17,19,22,26)(18,24,23,21)(20,27,28,32)(25,31,30,29)(33,36,39,46)(34,41,38,44)(35,43,37,45)(40,47,42,48);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,35,39,37)(34,40,38,42)(36,43,46,45)(41,47,44,48);(1,5,6,14)(2,8,9,13)(3,11,4,12)(7,15,10,16)(17,20,22,28)(18,25,23,30)(19,27,26,32)(21,29,24,31)(33,37,39,35)(34,42,38,40)(36,45,46,43)(41,48,44,47);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,21,22,24)(18,26,23,19)(20,29,28,31)(25,32,30,27)(33,38,39,34)(35,42,37,40)(36,41,46,44)(43,47,45,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8094112);