#9039403

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

3975 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)(3,8)(4,12)(5,7)(6,11)(9,13)(10,14)(15,16)(19,25)(23,26)(27,31)(30,32)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(1,3)(2,6)(4,10)(5,11)(7,8)(9,14)(12,15)(13,16)(17,18,21,22)(19,23,25,26)(20,24,28,29)(27,30,31,32)(36,44)(39,46)(42,48)(45,47);(2,7)(4,13)(6,8)(10,16)(18,22)(20,28)(23,26)(27,31)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,3)(2,8)(4,10)(5,11)(6,7)(9,15)(12,14)(13,16)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32);(1,2)(3,6)(4,9)(5,7)(8,11)(10,14)(12,13)(15,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47);(1,4,5,13)(2,9,7,12)(3,10,11,16)(6,14,8,15)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=i,e^2=gh,f^2=g,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, 8946599);