#10166250

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	#39322	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,d^2=h,e^2=ghi,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, 10073446);