#10480786

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
prime radical	#48458	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4142 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=i,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, 10387982);