#10047005

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

4268 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9954201);