#10247352

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

4117 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=ghi,e^2=gh,f^2=g,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, 10154548);