#9333323

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	#41291	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4002 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=ghi,e^2=h,f^2=gi,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, 9240519);