#9243225

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	#49477	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4117 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)(9,13)(10,16)(14,15)(17,18,22,25)(19,21,28,31)(20,24,30,23)(26,32,27,29)(33,34)(35,44)(36,46)(37,41)(38,45)(39,40)(42,47)(43,48);(3,9)(6,13)(10,15)(14,16)(17,19,22,28)(18,21,25,31)(20,27,30,26)(23,32,24,29)(34,40)(35,37)(42,46)(43,45);(1,2,5,8)(3,6,9,13)(4,7,12,11)(10,14,15,16)(17,20,22,30)(18,23,25,24)(19,26,28,27)(21,29,31,32)(33,35,39,37)(34,41,40,44)(36,43,47,45)(38,42,48,46);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,20)(18,24)(19,27)(21,29)(22,30)(23,25)(26,28)(31,32)(33,36,39,47)(34,42,40,46)(35,45,37,43)(38,44,48,41);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,19)(18,21)(20,27)(22,28)(23,32)(24,29)(25,31)(26,30)(33,37,39,35)(34,41,40,44)(36,43,47,45)(38,46,48,42);(1,4)(2,7)(3,10)(5,12)(6,14)(8,11)(9,15)(13,16)(17,21)(18,19)(20,29)(22,31)(23,26)(24,27)(25,28)(30,32)(33,38)(34,43)(35,42)(36,44)(37,46)(39,48)(40,45)(41,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=h,d^2=g,e^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, 9150421);