#10106949

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	#57795	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3914 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=ghi,d^2=h,e^2=g,f^2=g,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, 10014145);