#10482495

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

4234 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)(3,9,10,5)(4,8,15,7)(12,14,13,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,31)(28,29)(30,32)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(1,3)(2,5)(4,12)(6,10)(7,14)(8,16)(9,11)(13,15)(17,19,23,22)(18,24,20,26)(21,27,29,30)(25,31,28,32)(33,34)(35,36)(37,40)(38,41)(39,43)(42,44)(45,46)(47,48);(1,4)(2,7)(3,12)(5,14)(6,15)(8,11)(9,16)(10,13)(17,18,23,20)(19,26,22,24)(21,25,29,28)(27,32,30,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(1,4)(2,8)(3,13)(5,14)(6,15)(7,11)(9,16)(10,12)(17,20,23,18)(19,24,22,26)(21,25,29,28)(27,32,30,31)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,3)(2,9)(4,12)(5,11)(6,10)(7,16)(8,14)(13,15)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,47)(46,48);(1,5,6,9)(2,10,11,3)(4,14,15,16)(7,13,8,12)(17,22,23,19)(18,24,20,26)(21,27,29,30)(25,32,28,31)(33,37,38,46)(34,40,41,45)(35,42,43,48)(36,44,39,47)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=i,f^2=h,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, 10389691);