#9783320

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	#52324	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3980 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9690516);