#9368136

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

4342 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,d^2=ghi,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=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, 9275332);