#9240981

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	#54070	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3865 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)(3,11)(4,14)(5,15)(6,10)(7,13)(8,12)(9,16)(17,18,23,22)(19,26,21,24)(20,28,30,25)(27,31,29,32)(33,34,38,42)(35,39,43,48)(36,40,46,45)(37,41,47,44);(1,3)(2,7)(4,5)(6,13)(8,15)(9,14)(10,11)(12,16)(17,19,23,21)(18,24,22,26)(20,27,30,29)(25,31,28,32)(35,43)(37,47)(39,48)(41,44);(1,2,6,10)(3,7,13,11)(4,14,12,8)(5,9,16,15)(17,18,23,22)(19,24,21,26)(20,25,30,28)(27,31,29,32)(33,35)(34,39)(36,41)(37,45)(38,43)(40,47)(42,48)(44,46);(1,4,6,12)(2,8,10,14)(3,5,13,16)(7,15,11,9)(17,20)(18,25)(19,27)(21,29)(22,28)(23,30)(24,31)(26,32);(1,2)(3,11)(4,8)(5,9)(6,10)(7,13)(12,14)(15,16)(17,21,23,19)(18,24,22,26)(20,29,30,27)(25,31,28,32)(33,36)(34,40)(35,44)(37,39)(38,46)(41,43)(42,45)(47,48);(1,5)(2,9)(3,12)(4,13)(6,16)(7,8)(10,15)(11,14)(17,22,23,18)(19,24,21,26)(20,28,30,25)(27,31,29,32)(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=gh,c^2=g,d^2=h,e^2=gh,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, 9148177);