#9039180

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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,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, 8946376);