#10056600

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

4114 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32)(33,34,39,40)(35,38,36,43)(37,41,45,48)(42,47,44,46);(4,11)(8,14)(10,15)(12,16)(17,18,21,24)(19,22,25,29)(20,23,28,32)(26,30,27,31)(33,35)(34,38)(36,39)(37,44)(40,43)(41,46)(42,45)(47,48);(2,7)(3,9)(8,14)(10,15)(19,25)(22,29)(26,27)(30,31)(34,40)(35,36)(37,45)(46,47);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32)(33,36,39,35)(34,38,40,43)(37,44,45,42)(41,47,48,46);(1,4,6,11)(2,8,7,14)(3,10,9,15)(5,12,13,16)(17,19,21,25)(18,22,24,29)(20,27,28,26)(23,31,32,30)(33,37)(34,41)(35,42)(36,44)(38,47)(39,45)(40,48)(43,46);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,20,21,28)(18,23,24,32)(19,26,25,27)(22,30,29,31)(33,38)(34,35)(36,40)(37,46)(39,43)(41,44)(42,48)(45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=h,e^2=ghi,f^2=g,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, 9963796);