#10504972

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

4180 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10412168);