#10166340

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4068 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=hi,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, 10073536);