#10480915

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

4251 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)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(18,19)(21,29)(23,25)(27,30)(33,34)(35,42)(36,40)(37,45)(38,39)(41,46)(43,47)(44,48);(2,6)(7,13)(8,14)(15,16)(18,19)(23,25)(24,26)(31,32)(34,39)(40,42)(41,45)(47,48);(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,31,30,32)(33,35,38,36)(34,40,39,42)(37,43,46,44)(41,47,45,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,31,30,32)(33,34)(35,40)(36,42)(37,41)(38,39)(43,47)(44,48)(45,46);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(17,20)(18,23)(19,25)(21,30)(22,28)(24,32)(26,31)(27,29)(33,36,38,35)(34,40,39,42)(37,43,46,44)(41,48,45,47);(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,31)(25,32)(28,30)(33,37,38,46)(34,41,39,45)(35,43,36,44)(40,47,42,48)

Polycyclic presentation:

2^9|c^2=h,d^2=i,e^2=g,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, 10388111);