#10352403

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3826 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10259599);