#10340220

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

3972 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10247416);