#10143479

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	#55587	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4035 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gi,d^2=ghi,e^2=h,f^2=gh,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, 10050675);