#9240934

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	#42234	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3935 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)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18)(19,28)(20,21)(22,32)(23,25)(24,29)(26,31)(27,30)(33,34,38,42)(35,39,43,48)(36,40,46,45)(37,41,47,44);(2,6)(7,9)(8,13)(14,15)(17,19,23,29)(18,24,25,28)(20,30,31,22)(21,32,26,27)(35,43)(37,47)(39,48)(41,44);(4,12)(8,13)(10,16)(14,15)(18,25)(19,29)(20,31)(27,32)(33,35)(34,39)(36,41)(37,45)(38,43)(40,47)(42,48)(44,46);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,20)(18,26)(19,30)(21,25)(22,29)(23,31)(24,27)(28,32);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21,23,26)(18,20,25,31)(19,27,29,32)(22,28,30,24)(33,36)(34,40)(35,44)(37,39)(38,46)(41,43)(42,45)(47,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,22,23,30)(18,27,25,32)(19,20,29,31)(21,28,26,24)(33,37)(34,41)(35,45)(36,39)(38,47)(40,43)(42,44)(46,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=gh,e^2=h,f^2=h,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, 9148130);