#10047469

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

4034 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,32)(29,30)(33,34,39,45)(35,46,47,40)(36,42,48,41)(37,43,44,38);(1,3,6,11)(2,7,8,16)(4,14,10,12)(5,13,9,15)(19,20)(23,26)(27,28)(31,32)(33,35)(34,40)(36,38)(37,41)(39,47)(42,44)(43,48)(45,46);(2,8)(3,11)(4,10)(13,15)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(33,36,39,48)(34,41,45,42)(35,38,47,43)(37,46,44,40);(1,4,6,10)(2,9,8,5)(3,12,11,14)(7,13,16,15)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,36)(34,42)(35,43)(37,40)(38,47)(39,48)(41,45)(44,46);(1,3)(2,7)(4,12)(5,15)(6,11)(8,16)(9,13)(10,14)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,31)(28,32)(33,37)(34,43)(35,42)(36,40)(38,45)(39,44)(41,47)(46,48);(1,5,6,9)(2,10,8,4)(3,13,11,15)(7,12,16,14)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38,39,43)(34,44,45,37)(35,36,47,48)(40,42,46,41)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,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^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9954665);