#10504948

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

4009 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)(3,10)(4,7)(5,13)(6,9)(8,14)(11,15)(12,16)(17,18,22,26)(19,24,20,23)(21,25,30,29)(27,32,28,31)(33,34,39,36)(35,40,37,41)(38,42,46,44)(43,47,45,48);(19,20)(23,24)(27,28)(31,32)(34,36)(40,41)(42,44)(47,48);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,20,22,19)(18,24,26,23)(21,27,30,28)(25,31,29,32)(33,36,39,34)(35,41,37,40)(38,44,46,42)(43,48,45,47);(1,2)(3,7)(4,10)(5,8)(6,9)(11,15)(12,16)(13,14)(17,21,22,30)(18,25,26,29)(19,27,20,28)(23,31,24,32)(33,37,39,35)(34,41,36,40)(38,43,46,45)(42,47,44,48);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,18)(19,23)(20,24)(21,29)(22,26)(25,30)(27,32)(28,31)(33,38,39,46)(34,42,36,44)(35,43,37,45)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,d^2=gh,e^2=gi,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, 10412144);