#9039182

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

3971 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)(4,9)(5,10)(6,7)(11,13)(12,15)(14,16)(17,18)(19,24)(20,28)(21,26)(22,23)(25,29)(27,32)(30,31)(33,34,39,36)(35,42,41,38)(37,40,46,44)(43,48,47,45);(2,7)(4,13)(8,15)(10,14)(17,19,23,21)(18,24,22,26)(20,27,29,30)(25,31,28,32)(33,34,39,36)(35,38,41,42)(37,44,46,40)(43,48,47,45);(4,13)(5,16)(9,11)(10,14)(18,22)(19,21)(25,28)(27,30)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,3)(2,8)(4,14)(5,11)(6,12)(7,15)(9,16)(10,13)(17,20)(18,25)(19,27)(21,30)(22,28)(23,29)(24,31)(26,32)(33,36,39,34)(35,42,41,38)(37,44,46,40)(43,48,47,45);(1,4,6,13)(2,9,7,11)(3,10,12,14)(5,15,16,8)(17,21,23,19)(18,26,22,24)(20,27,29,30)(25,31,28,32)(33,37,39,46)(34,40,36,44)(35,43,41,47)(38,45,42,48);(1,5)(2,10)(3,11)(4,15)(6,16)(7,14)(8,13)(9,12)(17,22,23,18)(19,26,21,24)(20,28,29,25)(27,32,30,31)(33,38,39,42)(34,41,36,35)(37,45,46,48)(40,47,44,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=hi,e^2=gh,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, 8946378);