#9243715

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

4116 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,d^2=g,e^2=h,f^2=g,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, 9150911);