#10482471

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

4093 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=i,f^2=gh,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, 10389667);