#9243150

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	#57157	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4184 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,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, 9150346);