#10159195

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3848 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,15)(14,16)(17,18,22,26)(19,21,25,23)(20,24,30,28)(27,31,32,29)(33,34)(35,37)(36,40)(38,42)(39,46)(41,47)(43,48)(44,45);(2,7)(4,14)(8,11)(10,13)(17,19,22,25)(18,23,26,21)(20,27,30,32)(24,29,28,31)(33,35,38,41)(34,37,42,47)(36,44,46,43)(39,48,40,45);(2,7)(3,12)(9,16)(10,13)(33,36,38,46)(34,39,42,40)(35,43,41,44)(37,45,47,48);(1,3)(2,8)(4,10)(5,9)(6,12)(7,11)(13,14)(15,16)(17,18)(19,23)(20,28)(21,25)(22,26)(24,30)(27,31)(29,32)(33,36)(34,40)(35,44)(37,45)(38,46)(39,42)(41,43)(47,48);(1,4,6,14)(2,9,7,16)(3,13,12,10)(5,8,15,11)(17,20)(18,24)(19,27)(21,31)(22,30)(23,29)(25,32)(26,28)(33,35)(34,37)(36,43)(38,41)(39,45)(40,48)(42,47)(44,46);(1,5,6,15)(2,10,7,13)(3,9,12,16)(4,11,14,8)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,27,28,32)(33,37)(34,41)(35,42)(36,45)(38,47)(39,44)(40,43)(46,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,e^2=hi,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, 10066391);