#9039115

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

4107 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)(3,11)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(17,18,23,27)(19,28,20,24)(21,25,26,22)(29,30,31,32)(33,34)(35,38)(36,44)(37,41)(39,43)(40,45)(42,47)(46,48);(1,3,6,13)(2,7,10,11)(4,9,14,12)(5,8,16,15)(17,19,23,20)(18,24,27,28)(21,29,26,31)(22,30,25,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,47)(45,48);(4,14)(5,16)(8,15)(9,12)(17,19,23,20)(18,24,27,28)(21,31,26,29)(22,32,25,30)(33,34,37,41)(35,39,43,38)(36,44,45,40)(42,46,48,47);(1,2)(3,7)(4,15)(5,9)(6,10)(8,14)(11,13)(12,16)(17,20,23,19)(18,24,27,28)(21,29,26,31)(22,32,25,30)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,4,6,14)(2,8,10,15)(3,12,13,9)(5,11,16,7)(17,21,23,26)(18,25,27,22)(19,29,20,31)(24,32,28,30)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,45)(47,48);(1,5)(2,9)(3,8)(4,11)(6,16)(7,14)(10,12)(13,15)(17,22,23,25)(18,26,27,21)(19,30,20,32)(24,31,28,29)(33,36,37,45)(34,40,41,44)(35,42,43,48)(38,46,39,47)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8946311);