#10349115

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

4179 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=hi,d^2=h,e^2=gh,f^2=gh,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, 10256311);