#10090751

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

4271 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9997947);