#9240967

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
prime radical	#53840	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3840 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)(19,21)(20,28)(22,24)(23,27)(25,29)(26,31)(30,32)(33,34,38,42)(35,39,43,48)(36,40,46,45)(37,41,47,44);(2,5)(7,9)(8,13)(14,15)(17,19,22,27)(18,23,24,21)(20,26,29,32)(25,31,28,30)(35,43)(37,47)(39,48)(41,44);(4,12)(8,13)(10,16)(14,15)(18,24)(20,29)(21,23)(26,32)(33,35)(34,39)(36,41)(37,45)(38,43)(40,47)(42,48)(44,46);(1,3,6,11)(2,7,5,9)(4,10,12,16)(8,14,13,15)(17,20)(18,25)(19,26)(21,30)(22,29)(23,31)(24,28)(27,32);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,21,22,23)(18,19,24,27)(20,30,29,31)(25,26,28,32)(33,36)(34,40)(35,44)(37,39)(38,46)(41,43)(42,45)(47,48);(1,5,6,2)(3,9,11,7)(4,13,12,8)(10,15,16,14)(17,19)(18,23)(20,26)(21,24)(22,27)(25,31)(28,30)(29,32)(33,37)(34,41)(35,45)(36,39)(38,47)(40,43)(42,44)(46,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=gh,e^2=h,f^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, 9148163);