#9243703

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

4196 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,e^2=gh,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, 9150899);