#10349114

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

4218 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,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, 10256310);