#9240932

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3807 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)(3,6)(4,11)(5,8)(7,12)(9,15)(10,14)(13,16)(17,18,24,28)(19,21,22,25)(20,27,23,26)(29,30,32,31)(33,34,38,42)(35,39,43,48)(36,40,46,45)(37,41,47,44);(1,2,5,8)(3,6,10,14)(4,7,12,11)(9,13,16,15)(17,19)(18,25)(20,29)(21,28)(22,24)(23,32)(26,30)(27,31)(35,43)(37,47)(39,48)(41,44);(1,3,5,10)(2,6,8,14)(4,9,12,16)(7,13,11,15)(17,20,24,23)(18,26,28,27)(19,29,22,32)(21,31,25,30)(33,35)(34,39)(36,41)(37,45)(38,43)(40,47)(42,48)(44,46);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,16)(14,15)(17,21,24,25)(18,19,28,22)(20,30,23,31)(26,32,27,29);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,22,24,19)(18,25,28,21)(20,29,23,32)(26,31,27,30)(33,36)(34,40)(35,44)(37,39)(38,46)(41,43)(42,45)(47,48);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,23,24,20)(18,27,28,26)(19,29,22,32)(21,31,25,30)(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=gh,c^2=g,d^2=h,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, 9148128);