#10055745

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

4294 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,22,25)(19,20,23,26)(21,24,30,32)(27,28,31,29)(33,34)(35,42)(36,40)(37,41)(38,43)(39,45)(44,48)(46,47);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,32)(30,31)(33,35)(34,39)(36,44)(37,46)(38,45)(40,47)(41,48)(42,43);(2,3)(7,9)(8,10)(14,15)(19,23)(20,26)(27,31)(28,29)(33,36,38,37)(34,40,43,41)(35,44,45,46)(39,47,42,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,18)(19,26)(20,23)(21,24)(22,25)(27,29)(28,31)(30,32)(33,37,38,36)(34,41,43,40)(35,44,45,46)(39,47,42,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20,22,26)(18,23,25,19)(21,29,30,28)(24,27,32,31)(33,35)(34,42)(36,46)(37,44)(38,45)(39,43)(40,47)(41,48);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,32)(26,29)(33,34)(35,39)(36,40)(37,41)(38,43)(42,45)(44,47)(46,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=g,e^2=i,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, 9962941);