#10143415

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4011 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gi,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10050611);