#10439983

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

4231 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)(4,12)(5,9)(6,11)(8,13)(10,14)(15,16)(17,18)(19,25)(20,24)(21,23)(22,29)(26,30)(27,31)(28,32)(33,34,39,44)(35,42,37,40)(36,41,43,38)(45,46,47,48);(18,21)(24,25)(26,29)(31,32)(35,37)(40,42)(45,47)(46,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,35,39,37)(34,40,44,42)(36,45,43,47)(38,46,41,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,14)(13,16)(17,20,23,19)(18,25,21,24)(22,27,30,28)(26,31,29,32)(33,36)(34,41)(35,45)(37,47)(38,44)(39,43)(40,48)(42,46);(1,2)(3,6)(4,8)(5,9)(7,11)(10,14)(12,13)(15,16)(17,21,23,18)(19,25,20,24)(22,29,30,26)(27,32,28,31)(33,37,39,35)(34,42,44,40)(36,47,43,45)(38,48,41,46);(1,4,5,13)(2,8,9,12)(3,10,11,16)(6,14,7,15)(17,22,23,30)(18,26,21,29)(19,27,20,28)(24,31,25,32)(33,38,39,41)(34,43,44,36)(35,46,37,48)(40,47,42,45)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=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, 10347179);