#9723168

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	#49808	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4014 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9630364);