#10247045

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

4004 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,d^2=i,e^2=h,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, 10154241);