#9909999

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

4161 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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18,23,21)(19,25,20,24)(22,26,30,29)(27,32,28,31)(35,41)(39,45)(42,47)(46,48);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(34,38)(39,45)(40,43)(46,48);(2,7)(6,13)(8,11)(14,15)(17,19,23,20)(18,25,21,24)(22,27,30,28)(26,32,29,31)(33,34)(35,39)(36,40)(37,38)(41,45)(42,46)(43,44)(47,48);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,20,23,19)(18,24,21,25)(22,28,30,27)(26,31,29,32)(33,35)(34,39)(36,42)(37,41)(38,45)(40,46)(43,48)(44,47);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,21,23,18)(19,24,20,25)(22,26,30,29)(27,32,28,31)(33,34)(35,39)(36,43)(37,38)(40,44)(41,45)(42,48)(46,47);(1,4)(2,8)(3,9)(5,12)(6,14)(7,11)(10,16)(13,15)(17,22,23,30)(18,26,21,29)(19,27,20,28)(24,31,25,32)(33,36)(34,40)(35,42)(37,44)(38,43)(39,46)(41,47)(45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9817195);