#10246923

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

4050 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,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, 10154119);