#9783321

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	#54810	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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)(3,9)(4,6)(5,8)(7,11)(10,16)(12,14)(13,15)(18,24)(20,28)(25,30)(26,29)(33,34,39,43)(35,40,45,37)(36,46,44,41)(38,48,47,42);(1,3)(2,7)(4,9)(5,10)(6,11)(8,15)(12,16)(13,14)(17,18)(19,25)(20,22)(21,30)(23,24)(26,27)(28,32)(29,31)(33,35)(34,40)(36,47)(37,43)(38,44)(39,45)(41,48)(42,46);(2,4)(7,9)(8,12)(15,16)(17,19,23,21)(18,25,24,30)(20,26,28,29)(22,27,32,31)(33,36,39,44)(34,41,43,46)(35,38,45,47)(37,48,40,42);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,20)(18,22)(19,26)(21,29)(23,28)(24,32)(25,27)(30,31)(33,37,39,40)(34,35,43,45)(36,42,44,48)(38,41,47,46);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,21,23,19)(18,25,24,30)(20,29,28,26)(22,27,32,31)(33,35)(34,40)(36,38)(37,43)(39,45)(41,42)(44,47)(46,48);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,22)(18,20)(19,27)(21,31)(23,32)(24,28)(25,26)(29,30)(33,38)(34,42)(35,44)(36,45)(37,41)(39,47)(40,46)(43,48)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,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, 9690517);