#10166305

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	#56523	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3918 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=ghi,d^2=gh,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, 10073501);