#10106513

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

4071 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,d^2=g,e^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, 10013709);